Moments of Moments and Branching Random Walks.

We calculate, for a branching random walk X n ( l ) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable 1 2 n ∑ l = 1 2 n e 2 ß X n ( l ) , for ß ∈ R . We obtain explicit formulae for the first few moments for finite n. In the limit n → ∞ , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.

Random matrix theory and critical phenomena in quantum spin chains.

We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups U(N),O(N), and Sp(2N). In particular we calculate critical exponents s,ν, and z, corresponding to the energy gap, correlation length, and dynamic exponent, respectively. We also compute the ground state correlators 〈σ_{i}^{x}σ_{i+n}^{x}〉_{g},〈σ_{i}^{y}σ_{i+n}^{y}〉_{g}, and 〈∏_{i=1}^{n}σ_{i}^{z}〉_{g}, all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.

Number fields and function fields: coalescences, contrasts and emerging applications.

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained.

Quantum chaotic resonances from short periodic orbits.

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest-living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long-lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.

Quantum ergodicity on graphs.

We investigate the equidistribution of the eigenfunctions on quantum graphs in the high-energy limit. Our main result is an estimate of the deviations from equidistribution for large well-connected graphs. We use an exact field-theoretic expression in terms of a variant of the supersymmetric nonlinear sigma model. Our estimate is based on a saddle-point analysis of this expression and leads to a criterion for when equidistribution emerges asymptotically in the limit of large graphs. Our theory predicts a rate of convergence that is a significant refinement of previous estimates, long assumed to be valid for quantum chaotic systems, agreeing with them in some situations but not all. We discuss specific examples for which the theory is tested numerically.

Fractional variant Planck's over 2pi scaling for quantum kicked rotors without Cantori.

Previous studies of quantum delta-kicked rotors have found momentum probability distributions with a typical width (localization length L) characterized by fractional variant Planck's over 2pi scaling; i.e., L approximately variant Planck's over 2pi;{2/3} in regimes and phase-space regions close to "golden-ratio" cantori. In contrast, in typical chaotic regimes, the scaling is integer, L approximately variant Planck's over 2pi;{-1}. Here we consider a generic variant of the kicked rotor, the random-pair-kicked particle, obtained by randomizing the phases every second kick; it has no Kol'mogorov-Arnol'd-Moser mixed-phase-space structures, such as golden-ratio cantori, at all. Our unexpected finding is that, over comparable phase-space regions, it also has fractional scaling, but L approximately variant Planck's over 2pi;{-2/3}. A semiclassical analysis indicates that the variant Planck's over 2pi;{2/3} scaling here is of quantum origin and is not a signature of classical cantori.

Semiclassical structure of chaotic resonance eigenfunctions.

We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the nonunitary quantum propagator and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as variant Planck's over 2pi-->0 on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates [psi(variant Planck's over)] 2pi-->0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker's map, for which the probability density in position space is observed to have self-similarity properties.

Nodal domain statistics for quantum maps, percolation, and stochastic Loewner evolution.

We develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting statistical properties of the nodal domains is demonstrated for perturbed cat maps and supports the use of percolation theory to describe the wave functions of general Hamiltonian systems. We also demonstrate that the nodal domains of the perturbed cat maps obey the Cardy crossing formula and find evidence that the boundaries of the nodal domains are described by stochastic Loewner evolution with diffusion constant close to the expected value of 6, suggesting that quantum chaotic wave functions may exhibit conformal invariance in the semiclassical limit.

Entanglement in quantum spin chains, symmetry classes of random matrices, and conformal field theory.

We compute the entropy of entanglement between the first N spins and the rest of the system in the ground states of a general class of quantum spin chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like kappalog(2N+kappa as N-->infinity, where kappa and kappa are determined explicitly. In an important class of systems, kappa is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for kappa therefore provides an explicit formula for the central charge.

Intermediate wave function statistics.

We calculate statistical properties of the eigenfunctions of two quantum systems that exhibit intermediate spectral statistics: star graphs and Seba billiards. First, we show that these eigenfunctions are not quantum ergodic, and calculate the corresponding limit distribution. Second, we find that they can be strongly scarred, in the case of star graphs by short (unstable) periodic orbits and, in the case of Seba billiards, by certain families of orbits. We construct sequences of states which have such a limit. Our results are illustrated by numerical computations.

A multicentre retrospective audit of outcome of patients undergoing curative resection for rectal cancer.

AIM: This study describes the outcome of patients with rectal cancer treated in four New Zealand public hospitals before the advent of specialised colorectal units in order to provide a baseline against which any changes in management can be measured. METHODS: A retrospective review of case notes of patients who underwent resection of rectal cancer with curative intent over a period of 7-10 years up to 1995 in Christchurch, Wellington, Nelson, and Masterton Public Hospital's, was undertaken. Patients were identified from hospital records using a combination of methods (pathology data bases, clinical case mix data, operating logs and audit data). Metastatic disease was considered to be present if confirmed on histology or the clinical course of the patient was consistent with metastatic disease. Patients were excluded if there was perioperative evidence of metastatic disease or if they had transanal excision. Previously published results from Auckland and Dunedin Hospitals are compared. RESULTS: 524 patients with rectal cancer were identified who had undergone surgery with curative intent in the four hospitals. The overall permanent stoma rate was 37%. The overall 30-day mortality was 2.9%, five-year survival was 63% and local recurrence at five years was 26%. CONCLUSION: While low perioperative mortality and good long-term survival were achieved, there were high rates of local recurrence. These data are a baseline against which the impact of new approaches to curative resection for rectal cancer can be measured.

Use of ultrasonic surgical aspirator in operative cytoreduction of pseudomyxoma peritonei.

Pseudomyxoma peritonei presents a unique challenge to the surgical oncologist. Residual gelatinous tumor with varying degrees of adherence always remains on the abdominal viscera after standard excisional therapy. Traditionally, this has been removed by "electroevaporation" with ball-tip diathermy, but this is associated with an extensive peritoneal burn and associated ileus. We describe the use of an ultrasonic surgical aspirator as a safe and efficient method of tumor removal in this condition.

Complete genomic structure and mutational spectrum of PHKA2 in patients with x-linked liver glycogenosis type I and II.

X-linked liver glycogenosis (XLG) is probably the most frequent glycogen-storage disease. XLG can be divided into two subtypes: XLG I, with a deficiency in phosphorylase kinase (PHK) activity in peripheral blood cells and liver; and XLG II, with normal in vitro PHK activity in peripheral blood cells and with variable activity in liver. Both types of XLG are caused by mutations in the same gene, PHKA2, that encodes the regulatory alpha subunit of PHK. To facilitate mutation analysis in PHKA2, we determined its genomic structure. The gene consists of 33 exons, spanning >/=65 kb. By SSCP analysis of the different PHKA2 exons, we identified five new XLG I mutations, one new XLG II mutation, and one mutation present in both a patient with XLG I and a patient with XLG II, bringing the total to 19 XLG I and 12 XLG II mutations. Most XLG I mutations probably lead to truncation or disruption of the PHKA2 protein. In contrast, all XLG II mutations are missense mutations or small in-frame deletions and insertions. These results suggest that the biochemical differences between XLG I and XLG II might be due to the different nature of the disease-causing mutations in PHKA2. XLG I mutations may lead to absence of the alpha subunit, which causes an unstable PHK holoenzyme and deficient enzyme activity, whereas XLG II mutations may lead to in vivo deregulation of PHK, which might be difficult to demonstrate in vitro.

Vasculitis-induced colonic strictures: report of two cases.

PURPOSE: The clinical presentations of gastrointestinal involvement from systemic vasculitis are diverse. Colonic involvement from systemic vasculitis is unusual. We report the first case of a symptomatic colonic stricture associated with rheumatoid vasculitis and another associated with systemic lupus erythematosus. METHODS: The clinical, radiologic, and histologic features of two cases of symptomatic colonic strictures secondary to colonic involvement with vasculitis are described. The literature covering gastrointestinal involvement from vasculitis in these conditions is reviewed. RESULTS: Surgical resection of the colonic strictures was required in both patients and had a satisfactory outcome. CONCLUSIONS: These cases provide further evidence of the protean clinical presentations of intestinal involvement in systemic vasculitis. Although immunosuppression has been shown to be of value in the treatment of vasculitis affecting the gastrointestinal tract, surgical resection is required for established strictures.