On the Critical-Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective.

We study the 'critical moments' of subcritical Gaussian multiplicative chaos (GMCs) in dimensions . In particular, we establish a fully explicit formula for the leading order asymptotics, which is closely related to large deviation results for GMCs and demonstrates a similar universality feature. We conjecture that our result correctly describes the behaviour of analogous moments of moments of random matrices, or more generally structures which are asymptotically Gaussian and log-correlated in the entire mesoscopic scale.

A Spin Glass Model for the Loss Surfaces of Generative Adversarial Networks.

We present a novel mathematical model that seeks to capture the key design feature of generative adversarial networks (GANs). Our model consists of two interacting spin glasses, and we conduct an extensive theoretical analysis of the complexity of the model's critical points using techniques from Random Matrix Theory. The result is insights into the loss surfaces of large GANs that build upon prior insights for simpler networks, but also reveal new structure unique to this setting which explains the greater difficulty of training GANs.

Moments of Moments and Branching Random Walks.

We calculate, for a branching random walk to a leaf at depth on a binary tree, the positive integer moments of the random variable , for . We obtain explicit formulae for the first few moments for finite . In the limit , 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.

Combination of laser microdissection, 2D-DIGE and MALDI-TOF MS to identify protein biomarkers to predict colorectal cancer spread.

Biomarkers are urgently required to support current histological staging to provide additional accuracy in stratifying colorectal cancer (CRC) patients according to risk of spread to properly assign adjuvant chemotherapy after surgery. Chemotherapy is given to patients with stage III to reduce the risk of recurrence but is controversial in stage II patients. Up to 25% of stage II patients will relapse within 5 years after tumor removal and when this occurs cure is seldom possible.

The New Zealand PIPER Project: colorectal cancer survival according to rurality, ethnicity and socioeconomic deprivation-results from a retrospective cohort study.

Aim: To investigate differences in survival after diagnosis with colorectal cancer (CRC) by rurality, ethnicity and deprivation.

Methods: In this retrospective cohort study, clinical records and National Collections data were merged for all patients diagnosed with CRC in New Zealand in 2007-2008. Prioritised ethnicity was classified using New Zealand Cancer Registry data; meshblock of residence at diagnosis was used to determine rurality and socioeconomic deprivation.

Pair correlation and twin primes revisited.

We establish a connection between the conjectural two-over-two ratios formula for the Riemann zeta-function and a conjecture concerning correlations of a certain arithmetic function. Specifically, we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result. This casts a new light on the underpinnings of the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe.

Methods of a national colorectal cancer cohort study: the PIPER Project.

Aim: Colorectal cancer is one of the most common cancers, and second-leading cause of cancer-related death, in New Zealand. The PIPER (Presentations, Investigations, Pathways, Evaluation, Rx [treatment]) project was undertaken to compare presentation, investigations, management and outcomes by rurality, ethnicity and deprivation. This paper reports the methods of the project, a comparison of PIPER patient diagnoses to the New Zealand Cancer Registry (NZCR) data, and the characteristics of the PIPER cohort.

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.

Excision of the seminal vesicles for locally advanced and recurrent rectal and sigmoid cancer.

Background: This study aims to define the clinical and oncological outcome of 'en-bloc' excision of the seminal vesicles for locally advanced and recurrent tumours of the sigmoid and rectum.

Methods: Eight patients were identified from a prospective colorectal cancer database at a tertiary centre as having undergone excision of the seminal vesicles in continuity with a locally advanced or recurrent sigmoid or rectal adenocarcinoma. The presentation, operative details, histopathology, oncological outcome and morbidity of the procedure were assessed.

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.

Moments of zeta and correlations of divisor-sums: I.

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously, this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.

Treating pseudomyxoma peritonei without heated intraperitoneal chemotherapy--a first look in New Zealand.

Background: Pseudomyxoma peritonei is a condition characterised by dissemination of mucin-producing neoplastic cells throughout the peritoneal cavity. There are two pathological subsets, disseminated peritoneal adenomucinosis and peritoneal mucinosis carcinomatosis. Once a lethal disease, cytoreductive surgery combined with heated intraperitoneal chemotherapy (HIPEC) is challenging debulking as the standard of care.

Freezing transitions and extreme values: random matrix theory, ζ (1/2 + it) and disordered landscapes.

We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N → ∞.

Overexpression of aminoacylase 1 is associated with colorectal cancer progression.

Aminoacylase 1 (ACY1) is a cytosolic enzyme responsible for amino acid deacylation during intracellular protein degradation. ACY1 has been implicated in a number of human tumor types. However, the exact role of ACY1 in tumor development remains elusive because it was found to be lost in small cell lung cancer and renal cell carcinoma but overexpressed in colorectal cancer (CRC).

Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N×N random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function ζ(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random-matrix theory, and the theory of the Riemann zeta function.

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.

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.

A study into external rectal anatomy: improving patient selection for radiotherapy for rectal cancer.

Purpose: This study was designed to determine the distance from the anal verge to the anterior peritoneal reflection in vivo, thereby improving the selection of patients for preoperative radiotherapy.

Methods: Measurement of the distance from the anal verge to the anterior peritoneal reflection, confluence of the taenia, and the origin of the sigmoid mesentery in 50 patients in the lithotomy position.

Results: The mean distance from the anal verge to the anterior peritoneal reflection was 11.

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.

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.

CT manifestations of ileal dysgenesis.

Ileal dysgenesis is an uncommon condition of unknown etiology occurring in the distal ileum in the region of the vitelline duct. The CT appearance of this lesion, although not previously described to our knowledge, is characteristic. We report a patient with ileal dysgenesis who had an abdominal CT scan to evaluate chronic iron deficiency anemia and protein-losing enteropathy.

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.