Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operatorsThe fundamental elements of the variable separation theory are revisited, including the Eisenhart and Robertson theorems, Kalnins-Miller theory, and the intrinsic characterization of the separation of the Hamilton-Jacobi equation, in a unitary and geometrical perspective. The general notion of complete integrability of first-order normal systems of PDEs leads in a natural way to completeness conditions for separated solutions of the Schrödinger equation and to the Robertson condition. Two general types of multiplicative separation for the Schrödinger equation are defined and analyzed: they are called ''free'' and ''reduced'' separation, respectively. In the free separation the coordinates are necessarily orthogonal, while the reduced separation may occur in nonorthogonal coordinates, but only in the presence of symmetries ͑Killing vectors͒.Proof: By setting ⌫ a ϭ⌫ a Ϫ2g a␣ ␣ , V ϭVϩ⌫ ␣ ␣ Ϫg ␣␤ ␣ ␤ , 5201

The commutation relations of the first-order and second-order operators associated with the first integrals in involution of a Hamiltonian separable system are examined. It is shown that these operators commute if and only if a “pre-Robertson condition” is satisfied. This condition involves the Ricci tensor of the configuration manifold and it is implied by the Robertson condition, which is necessary and sufficient for the separability of the Schrödinger equation.

Epidemiological data on the impact of hypertensive crises (emergencies and urgencies) on referral to the Emergency Departments (EDs) are lacking, in spite of the evidence that they may be life-threatening conditions. We performed a multicenter study to identify all patients aged 18 years and over who were admitted to 10 Italian EDs during 2009 for hypertensive crises (systolic blood pressure ≥220 mmHg and/or diastolic blood pressure ≥120 mmHg). We classified patients as affected by either hypertensive emergencies or hypertensive urgencies depending on the presence or the absence of progressive target organ damage, respectively. Logistic regression analysis was then performed to assess variables independently associated with hypertensive emergencies with respect to hypertensive urgencies. Of 333,407 patients admitted to the EDs over the one-year period, 1,546 had hypertensive crises (4.6/1,000, 95% CI 4.4–4.9), and 23% of them had unknown hypertension. Hypertensive emergencies (n = 391, 25.3% of hypertensive crises) were acute pulmonary edema (30.9%), stroke (22.0%,), myocardial infarction (17.9%), acute aortic dissection (7.9%), acute renal failure (5.9%) and hypertensive encephalopathy (4.9%). Men had higher frequency than women of unknown hypertension (27.9% vs 18.5%, p<0.001). Even among known hypertensive patients, a larger proportion of men than women reported not taking anti-hypertensive drug (12.6% among men and 9.4% among women (p<0.001). Compared to women of similar age, men had higher likelihood of having hypertensive emergencies than urgencies (OR = 1.34, 95% CI 1.06–1.70), independently of presenting symptoms, creatinine, smoking habit and known hypertension. This study shows that hypertensive crises involved almost 5 out of 1,000 patients-year admitted to EDs. Sex differences in frequencies of unknown hypertension, compliance to treatment and risk of hypertensive emergencies might have implications for public health programs.

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 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 quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal multiseparability is discussed and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other threebody interactions widely studied in mathematical physics.

The additive variable separation in the Hamilton–Jacobi equation is studied for a natural Hamiltonian with scalar and vector potentials on a Riemannian manifold with positive–definite metric. The separation of this Hamiltonian is related to the separation of a suitable geodesic Hamiltonian over an extended Riemannian manifold. Thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of Killing webs, Killing tensors, and Killing vectors. The results are applicable to the case of a nondegenarate separation on a manifold with indefinite metric, where no null essential separable coordinates occur.