Symplectically-invariant soliton equations from non-stretching geometric curve flows

A moving frame formulation of geometric non-stretching flows of curves in the Riemannian symmetric spaces $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ is used to derive two bi-Hamiltonian hierarchies of symplectically-invariant soliton equations. As main results, multi-component versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting $Sp(1)\times Sp(n-1)$ invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schr\"odinger map.Comment: 39 pages; remarks added on algebraic aspects of the moving frame used in the constructio

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans

Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.Comment: 26 pages, 9 figure

Delay in referral of oropharyngeal squamous cell carcinoma to secondary care correlates with a more advanced stage at presentation, and is associated with poorer survival

Squamous carcinoma of the oropharynx presents with symptoms common to many benign diseases, and this can cause delay in referral to secondary care. We investigate delay in referral, defining this as the time from symptom-onset to date of general practitioners referral letter to secondary care, and the effect of that delay, using a retrospective case notes based study of patients presenting at our institution with oropharyngeal squamous carcinoma between 1995 and 2005. Using correlation analysis and ordinal regression, we examined the relationship between increased referral delay from primary care, clinical stage at presentation, and survival. Increasing time from symptom onset to referral to secondary care was positively correlated with more advanced disease stage at presentation (rs=+0.346, P=0.004). This was confirmed with ordinal regression modelling (delay estimate=0.045, P=0.042). Patients with delay of less than 6 weeks had significantly improved survival compared to those with a delay of greater than 6 weeks (P=0.032). For every 1 week of delay in referral, we estimate that the stage of presentation will progress by 0.045 of ‘a stage'

Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space $HP^n$. The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces $M=G/H$ by viewing $HP^n \simeq {\rm U}(n+1,H)/{\rm U}(1,H) \times {\rm U}(n,H)\simeq {\rm Sp}(n+1)/{\rm Sp}(1)\times {\rm Sp}(n)$ as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar-vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $HP^n$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrodinger map.Comment: 25 pages; typos correcte

Sexual Size Dimorphism and Body Condition in the Australasian Gannet

Funding: The research was financially supported by the Holsworth Wildlife Research Endowment. Acknowledgments We thank the Victorian Marine Science Consortium, Sea All Dolphin Swim, Parks Victoria, and the Point Danger Management Committee for logistical support. We are grateful for the assistance of the many field volunteers involved in the study.Peer reviewedPublisher PD

Sialadenosis in Patients with Advanced Liver Disease

Sialadenosis (sialosis) has been associated most often with alcoholic liver disease and alcoholic cirrhosis, but a number of nutritional deficiencies, diabetes, and bulimia have also been reported to result in sialadenosis. The aim of this study was to determine the prevalence of sialadenosis in patients with advanced liver disease. Patients in the study group consisted of 300 candidates for liver transplantation. Types of liver disease in subjects with clinical evidence of sialadenosis were compared with diagnoses in cases who had no manifestations of sialadenosis. The data were analyzed for significant association. Sialadenosis was found in 28 of the 300 subjects (9.3%). Among these 28 cases, 11 (39.3%) had alcoholic cirrhosis. The remaining 17 (60.7%) had eight other types of liver disease. There was no significant association between sialadenosis and alcoholic cirrhosis (P = 0.389). These findings suggest that both alcoholic and non-alcoholic cirrhosis may lead to the development of sialadenosis. Advanced liver disease is accompanied by multiple nutritional deficiencies which may be exacerbated by alcohol. Similar metabolic abnormalities may occur in patients with diabetes or bulimia. Malnutrition has been associated with autonomic neuropathy, the pathogenic mechanism that has been proposed for sialadenosis

Oral cancer treatment costs in Greece and the effect of advanced disease

BACKGROUND: The main purpose of the study was to quantify the direct costs of oral cancer treatment to the healthcare system of Greece. Another aim was to identify factors that affect costs and potential cost reduction items. More specifically, we examined the relationship between stage of disease, modality of treatment and total direct costs. METHODS: The medical records and clinic files of the Oral and Maxillofacial Clinic of the Athens General Hospital "Genimatas" were abstracted to investigate clinical treatment characteristics, including length of hospitalization, modes of treatment, stage of disease etc. Records of 95 patients with oral squamous cell carcinoma (OSSC), with at least six months of follow-up, were examined. The clinical data was then used to calculate actual direct costs, based on 2001 market values. RESULTS: The mean total direct costs for OSSC treatment estimated at euro 8,450 or approximately US$ 7,450. Costs depended on the stage of the disease, with significant increases in stages III and IV, as compared with stages I and II (p < 0.05). Multi-modality treatment applied mainly to patients in stages III and IV was the factor that affected the cost. Disease stage was also associated with the total duration of hospitalization (p < 0.05). CONCLUSIONS: The clinical management of advanced oral cancer is strongly associated with higher costs. Although the ideal would be to prevent cancer, the combination of high-risk screening, early diagnosis and early treatment seems the most efficient way to reduce costs, and most importantly, prolong life

Periodontitis is associated with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease

This work was funded by the Diabetes Research and Wellness Fund. WA is in receipt of an MRC New Investigator Award