First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians $H$ obtained as one-dimensional extensions of natural (geodesic) $n$-dimensional Hamiltonians $L$. The Liouville integrability of $L$ implies the (minimal) superintegrability of $H$. We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with $L$ is constant. As examples, the procedure is applied to one-dimensional $L$, including and improving earlier results, and to two and three-dimensional $L$, providing new superintegrable systems.Comment: Theorem 1, Lemmas 1 and 2, Example 2 are correcte

Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. St\"ackel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given.Comment: 18 page

Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization

In previous papers we determined necessary and sufficient conditions for the existence of a class of natural Hamiltonians with non-trivial first integrals of arbitrarily high degree in the momenta. Such Hamiltonians were characterized as (n+1)-dimensional extensions of n-dimensional Hamiltonians on constant-curvature (pseudo-)Riemannian manifolds Q. In this paper, we generalize that approach in various directions, we obtain an explicit expression for the first integrals, holding on the more general case of Hamiltonians on Poisson manifolds, and show how the construction of above is made possible by the existence on Q of particular conformal Killing tensors or, equivalently, particular conformal master symmetries of the geodesic equations. Finally, we consider the problem of Laplace-Beltrami quantization of these first integrals when they are of second-degree.Comment: Presented at the conference Quantum Theory and Symmetries 7, Praha, August 7-13 2011. In v2 some typos corrected, a comment added after eq. (4), a comment about ref. [1] correcte

Superintegrable 3-body systems on the line

We consider classical three-body interactions on a Euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momenta first integrals. These systems are superseparable (i.e. multiseparable), superintegrable and equivalent (up to rescalings) to a one-particle system in the three-dimensional Euclidean space. Common features of the dynamics are discussed. We show how to determine the quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal superseparability is proved and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other three-body interactions widely studied in mathematical physics.Comment: Corrected typos. Some improvement

Effectiveness of an image analyzing AI-based Digital Health Technology to identify Non-Melanoma Skin Cancer and other skin lesions: results of the DERM-003 study

IntroductionIdentification of skin cancer by an Artificial Intelligence (AI)-based Digital Health Technology could help improve the triage and management of suspicious skin lesions.MethodsThe DERM-003 study (NCT04116983) was a prospective, multi-center, single-arm, masked study that aimed to demonstrate the effectiveness of an AI as a Medical Device (AIaMD) to identify Squamous Cell Carcinoma (SCC), Basal Cell Carcinoma (BCC), pre-malignant and benign lesions from dermoscopic images of suspicious skin lesions. Suspicious skin lesions that were suitable for photography were photographed with 3 smartphone cameras (iPhone 6S, iPhone 11, Samsung 10) with a DL1 dermoscopic lens attachment. Dermatologists provided clinical diagnoses and histopathology results were obtained for biopsied lesions. Each image was assessed by the AIaMD and the output compared to the ground truth diagnosis.Results572 patients (49.5% female, mean age 68.5 years, 96.9% Fitzpatrick skin types I-III) were recruited from 4 UK NHS Trusts, providing images of 611 suspicious lesions. 395 (64.6%) lesions were biopsied; 47 (11%) were diagnosed as SCC and 184 (44%) as BCC. The AIaMD AUROC on images taken by iPhone 6S was 0.88 (95% CI: 0.83–0.93) for SCC and 0.87 (95% CI: 0.84–0.91) for BCC. For Samsung 10 the AUROCs were 0.85 (95% CI: 0.79–0.90) and 0.87 (95% CI, 0.83–0.90), and for the iPhone 11 they were 0.88 (95% CI, 0.84–0.93) and 0.89 (95% CI, 0.86–0.92) for SCC and BCC, respectively. Using pre-determined diagnostic thresholds on images taken on the iPhone 6S the AIaMD achieved a sensitivity and specificity of 98% (95% CI, 88–100%) and 38% (95% CI, 33–44%) for SCC; and 94% (95% CI, 90–97%) and 28% (95 CI, 21–35%) for BCC. All 16 lesions diagnosed as melanoma in the study were correctly classified by the AIaMD.DiscussionThe AIaMD has the potential to support the timely diagnosis of malignant and premalignant skin lesions

Systematic versus on-demand early palliative care: results from a multicentre, randomised clinical trial

Background Early palliative care (EPC) in oncology has been shown to have a positive impact on clinical outcome, quality-of-care outcomes, and costs. However, the optimal way for activating EPC has yet to be defined. Methods This prospective, multicentre, randomised study was conducted on 207 outpatients with metastatic or locally advanced inoperable pancreatic cancer. Patients were randomised to receive ‘standard cancer care plus on-demand EPC’ (n = 100) or ‘standard cancer care plus systematic EPC’ (n = 107). Primary outcome was change in quality of life (QoL) evaluated through the Functional Assessment of Cancer Therapy – Hepatobiliary questionnaire between baseline (T0) and after 12 weeks (T1), in particular the integration of physical, functional, and Hepatic Cancer Subscale (HCS) combined in the Trial Outcome Index (TOI). Patient mood, survival, relatives' satisfaction with care, and indicators of aggressiveness of care were also evaluated. Findings The mean changes in TOI score and HCS score between T0 and T1 were −4.47 and −0.63, with a difference between groups of 3.83 (95% confidence interval [CI] 0.10–7.57) (p = 0.041), and −2.23 and 0.28 (difference between groups of 2.51, 95% CI 0.40–4.61, p = 0.013), in favour of interventional group. QoL scores at T1 of TOI scale and HCS were 84.4 versus 78.1 (p = 0.022) and 52.0 versus 48.2 (p = 0.008), respectively, for interventional and standard arm. Until February 2016, 143 (76.9%) of the 186 evaluable patients had died. There was no difference in overall survival between treatment arms. Interpretations Systematic EPC in advanced pancreatic cancer patients significantly improved QoL with respect to on-demand EPC