Local mode and normal mode interpretations of the CH and CD stretching vibrational manifolds of 1,1-dideuteroethene

The =CH2 AND =CD2 stretching vibrational overtones of H2C=CD2 have been studied up to V= 6 and V= 3, respectively. We report their interpretation in terms of a transition from normal to local modes, involving Fermi resonance with the C=C stretching and CH2 scissoring vibrations. We discuss the alternative representation of the vibrational Hamiltonian matrix in local mode and normal mode basis functions, and conclude that the normal mode basis offers greater flexibility in representing small anharmonic couplings with other modes.

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.

The Brownian traveller on manifolds

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

Carbon suboxide: the vibrational dependence of the v7 bending potential function

Data on the vibrational energy levels and rotational constants of carbon suboxide for the low-wavenumber bending mode ν7 are reviewed, in the ground-state manifold, and in the ν2-, ν3-, ν4-, and ν2 + ν4-state manifolds. Following the procedure developed by Duckett, Mills, and Robiette [J. Mol. Spectrosc. 63, 249 (1976)] the data have been inverted to give the effective bending potential in ν7 for each of these five states. Values are obtained for various other parameters in the effective vibration-rotation Hamiltonian. The potential and rotational constants in ν2 + ν4 are given to a close approximation by linear extrapolation from the ground state through the ν2 and ν4 states.

The geometry of optimal control solutions on some six dimensional lie groups

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E-3, the spheres S-3 and the hyperboloids H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E³, the spheres S³ and the hyperboloids H³ with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions is illustrated.

Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

Studying reactive processes with classical dynamics: rebinding dynamics in MbNO

A new surface-crossing algorithm suitable for describing bond-breaking and bond-forming processes in molecular dynamics simulations is presented. The method is formulated for two intersecting potential energy manifolds which dissociate to different adiabatic states. During simulations, crossings are detected by monitoring an energy criterion. If fulfilled, the two manifolds are mixed over a finite number of time steps, after which the system is propagated on the second adiabat and the crossing is carried out with probability one. The algorithm is extensively tested (almost 0.5 mu s of total simulation time) for the rebinding of NO to myoglobin. The unbound surface ((FeNO)-N-...) is represented using a standard force field, whereas the bound surface (Fe-NO) is described by an ab initio potential energy surface. The rebinding is found to be nonexponential in time, in agreement with experimental studies, and can be described using two time constants. Depending on the asymptotic energy separation between the manifolds, the short rebinding timescale is between 1 and 9 ps, whereas the longer timescale is about an order of magnitude larger. NO molecules which do not rebind within 1 ns are typically found in the Xenon-4 pocket, indicating the high affinity of NO to this region in the protein.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

Spectral analysis and the Aharonov-Bohm\~ effect on certain almost-Riemannian manifold

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

Compensation and Rewards for Environmental Services (CRES) and efficient design of contracts in developing countries. Behavioral insights from a natural field experiment

The use of economic incentives for biodiversity (mostly Compensation and Reward for Environmental Services including Payment for ES) has been widely supported in the past decades and became the main innovative policy tools for biodiversity conservation worldwide. These policy tools are often based on the insight that rational actors perfectly weigh the costs and benefits of adopting certain behaviors and well-crafted economic incentives and disincentives will lead to socially desirable development scenarios. This rationalist mode of thought has provided interesting insights and results, but it also misestimates the context by which ‘real individuals’ come to decisions, and the multitude of factors influencing development sequences. In this study, our goal is to examine how these policies can take advantage of some unintended behavioral reactions that might in return impact, either positively or negatively, general policy performances. We test the effect of income's origin (‘Low effort’ based money vs. ‘High effort’ based money) on spending decisions (Necessity vs. Superior goods) and subsequent pro social preferences (Future pro-environmental behavior) within Madagascar rural areas, using a natural field experiment. Our results show that money obtained under low effort leads to different consumption patterns than money obtained under high efforts: superior goods are more salient in the case of low effort money. In parallel, money obtained under low effort leads to subsequent higher pro social behavior. Compensation and rewards policies for ecosystem services may mobilize knowledge on behavioral biases to improve their design and foster positive spillovers on their development goals.