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

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

Determinants of DNA yield and purity collected with buccal cell samples

Buccal cells are an important source of DNA in epidemiological studies, but little is known about factors that influence amount and purity of DNA. We assessed these factors in a self-administered buccal cell collection procedure, obtained with three cotton swabs. In 2,451 patients DNA yield and in 1,033 patients DNA purity was assessed. Total DNA yield ranged from 0.08 to 1078.0 μg (median 54.3 μg; mean 82.2 μg ± SD 92.6). The median UV 260:280 ratio, was 1.95. Samples from men yielded significantly more DNA (median 58.7 μg) than those from women (median 44.2 μg). Diuretic drug users had significantly lower purity (median 1.92) compared to other antihypertensive drug users (1.95). One technician obtained significantly lower DNA yields. Older age was associated with lower DNA purity. In conclusion, DNA yield from buccal swabs was higher in men and DNA purity was associated with age and the use of diuretics

Diabetes mellitus and oral lichen planus: A systematic review and meta-analysis

Objective: To undertake a meta-analysis of the association of Oral Lichen Planus (OLP) with diabetes, two diseases with an important impact on public health and the economy, but the evidence of which about their association is inconsistent. Methods: Relevant studies were localized by searching MEDLINE, EMBASE, Conference Proceedings, and other databases from inception to October 2020, without restrictions. The reference lists of included studies and of related reviews were also inspected. Global pooled odds ratios were calculated, and predefined subgroup analyses were performed. The heterogeneity between studies and publication bias was assessed and sensitivity analysis was carried out. Results: Thirty-two studies were included in the meta-analysis. Pooled ORs showed a moderate association between diabetes and OLP [OR: 1.87 (95%CI: 1.57, 2.34)]. The association is limited to studies carried out on adults only [OR: 2.12 (95%CI: 1.75, 2.57)] and is observed in all study designs. Globally, the heterogeneity was low to moderate. Studies carried out in European populations show a stronger association of diabetes and OLP than Asiatic studies [OR: 2.49 (95%CI: 1.87, 3.32) and 1.60 (95%CI: 1.25, 2.03), respectively]. Conclusions: Diabetes and OLP are moderately associated. Systematic diagnosis of diabetes in OLP patients could prove usefulS