Geodesics on Lie groups: Euler equations and totally geodesic subgroup

The geodesic motion on a Lie group equipped with a left or right invariant Riemannian metric is governed by the Euler-Arnold equation. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. Results on the construction and characterisation of such metrics are given. The setting works both in the classical nite dimensional case, and in the category of in nite dimensional Fr echet Lie groups, in which di eomorphism groups are included. Using the framework we give new examples of both nite and in nite dimensional totally geodesic subgroups. In particular, based on the cross helicity, we construct right invariant metrics such that a given subgroup of exact volume preserving di eomorphisms is totally geodesic. The paper also gives a general framework for the representation of Euler-Arnold equations in arbitrary choice of dual pairing

Geodesic Warps by Conformal Mappings

In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications e.g.\ in medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern methods, which are drawn from the full set of diffeomorphism, he deliberately chose lower-dimensional sets of transformations, such as planar conformal mappings. In this paper we study warps of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several examples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations

Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $\epsilon$, and the schemes under study preserve the symplectic structure in the case $\epsilon=0$. In the case $0<\epsilon\ll 1$ the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3--D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order

Diffeomorphic random sampling using optimal information transport

In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)---an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge--Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.Comment: 8 pages, 3 figure

Geometric Generalisations of SHAKE and RATTLE

A geometric analysis of the Shake and Rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises Shake and Rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting. In order for Shake and Rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given

Middleborns disadvantaged? testing birth-order effects on fitness in pre-industrial finns

Parental investment is a limited resource for which offspring compete in order to increase their own survival and reproductive success. However, parents might be selected to influence the outcome of sibling competition through differential investment. While evidence for this is widespread in egg-laying species, whether or not this may also be the case in viviparous species is more difficult to determine. We use pre-industrial Finns as our model system and an equal investment model as our null hypothesis, which predicts that (all else being equal) middleborns should be disadvantaged through competition. We found no overall evidence to suggest that middleborns in a family are disadvantaged in terms of their survival, age at first reproduction or lifetime reproductive success. However, when considering birth-order only among same-sexed siblings, first-, middle-and lastborn sons significantly differed in the number of offspring they were able to rear to adulthood, although there was no similar effect among females. Middleborn sons appeared to produce significantly less offspring than first-or lastborn sons, but they did not significantly differ from lastborn sons in the number of offspring reared to adulthood. Our results thus show that taking sex differences into account is important when modelling birth-order effects. We found clear evidence of firstborn sons being advantaged over other sons in the family, and over firstborn daughters. Therefore, our results suggest that parents invest differentially in their offspring in order to both preferentially favour particular offspring or reduce offspring inequalities arising from sibling competition

Elucidation of the liquid-liquid distribution behavior of ion associates of metal-halogeno complex anions with quaternary ammonium counter cations and its application to separation and analysis

第四級アンモニウムイオンを対イオンとする一価, 二価金属ハロゲノ錯陰イオンのイオン会合抽出挙動を把握し, 分離・分析的応用を図るために, 炭素数及び形状の異なる第四級アンモニウム陽イオンを用いて, 水-各種抽出溶媒 {1,2-ジクロロエタン, クロロホルム (CF), クロロベンゼン (Cl-B), ベンゼン (B), トルエン (T) 及び四塩化炭素 (CTC)} 系での抽出定数を求めた. 得られた抽出定数から次の知見を得た. (1) 配位子の抽出性に及ぼす影響 : 配位子がCl-, Br-, I-と変わるにつれ, この順に抽出性は良くなる. (2) 配位子数の影響 : 配位子数が多くなるに従い, 抽出性も良くなる. (3) 中心金属の影響 : 配位子数が同じ場合には, 抽出性はほぼ中心金属イオンの大きさの順となる. (4) 金属錯陰イオンの電荷の影響 : 一般に二価陰イオンよりも一価陰イオンのほうが抽出されやすい. (5) 対陽イオンのアルキル鎖のメチレン基の寄与 : メチレン基一つ当たり, 大体0.4～0.8の抽出定数 (log K(ex)) の増大となる. (6) 抽出溶媒の影響 : 抽出溶媒の抽出能は次の順となる : CTC<T<B<Cl-B<CF. (7) 金属ハロゲノ錯陰イオンの配位子の違いによる抽出定数の差 (Δlog K(ex)) は溶媒によらず, ほぼ一定である. これらの知見を基に, 金属ハロゲノ錯陰イオンと疎水性陽イオンとのイオン会合抽出を利用する幾つかの金属の分離・定量法を開発した.The distribution behavior of ion associates of both monovalent and divalent metal-halogeno complex anions with various quaternary ammonium cations between the aqueous phase and several organic phases {1,2-dichloroethane, chloroform (CF), chlorobenzene (Cl-B), benzene (B), toluene (T) and carbon tetrachloride (CTC)} was examined, and the extraction constants (log Kex) were determined. The larger is the size of the ligand (Cl-<Br-<I-) and the coordination number, the greater is the ion associability. For the same coordination number, in general, the larger is the size of the metal ion, the greater is the ion associability. In general, the extractability of monovalent metal-halogeno complex anions is larger than that of divalent metal-halogeno complex anions. A linear relationship was obtained between log Kex and the number of carbon atoms in quaternary ammonium ion, and the contribution of a methylene group to the extraction constant (Δlog K(ex)/-CH(2-)) was found to be about 0.4∼0.8. Among the ion associates examined, the order of the extractability of the extracting solvent was generally CTC<T<B<Cl-B<CF. Also, the order of the extractability of the ion associates for dihalogenocuprate (I), tetrahalogenoaurate (III) and tetrahalogenothallate (III) complex ions was as follows, respectively : CuCl(2)-<CuBr(2)-<CuI(2)- ; AuCl(4)-<AuBr(4)- ; TlCl(4)-<TlBr(4)-<TlI(4-). The values of Δlog K(ex) between the complex anions were almost equal, even though the extracting solvents were changed. From these results, several extraction-spectrophotometric methods for the determination of metal based on the formation of an ion associate of metal-halogeno complex anion with hydrophobic cations were developed

A Case Series on Genotype and Outcome of Liver Transplantation in Children with Niemann-Pick Disease Type C

Background: To report on clinical presentation and outcomes of children who underwent liver transplantation (LTx) and were subsequently diagnosed to have Niemann-Pick type C (NPC). Methods: Retrospective, descriptive, multi-centre review of children diagnosed with NPC who underwent LTx (2003–2018). Diagnosis was made by filipin skin test or genetic testing. Results: Nine children were identified (six centres). Neonatal acute liver failure was the most common indication for LTx (seven children). Median age at first presentation: 7 days (range: 0–37). The most prevalent presenting symptoms: jaundice (8/9), hepatosplenomegaly (8/9) and ascites (6/9). 8/9 children had a LTx before the diagnosis of NPC. Genetic testing revealed mutations in NPC1 correlating with a severe biochemical phenotype in 5 patients. All 9 children survived beyond early infancy. Seven children are still alive (median follow-up time of 9 (range: 6–13) years). Neurological symptoms developed in 4/7 (57%) patients at median 9 (range: 5–13) years following LTx. Conclusion: Early diagnosis of NPC continues to be a challenge and a definitive diagnosis is often made only after LTx. Neurological disease is not prevented in the majority of patients. Genotype does not appear to predict neurological outcome after LTx. LTx still remains controversial in NPC