Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds

The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities

Neighborhoods and manifolds of immersed curves

We present some fine properties of immersions ℐ:M N between manifolds, with particular attention to the case of immersed curves c:S1 ℝn. We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, neighborhoods of immersed and freely immersed curve, and local unique representations of nearby such curves, possibly "up to reparameterization."We present examples and counterexamples to support the significance of these results. Eventually, we provide a complete and detailed proof of a result first stated in a 1991-paper by Cervera, Mascaró, and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold

An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

Here shape space is either the manifold of simple closed smooth unparameterized curves in $\mathbb R^2$ or is the orbifold of immersions from $S^1$ to $\mathbb R^2$ modulo the group of diffeomorphisms of $S^1$. We investige several Riemannian metrics on shape space: $L^2$-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order $n$ on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on $\mathbb R^2$is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.Comment: 46 pages, some misprints correcte

Influencia de las variables de ensayo en la conductividad térmica de hormigones aislantes refractarios

Se realizaron experiencias en base a los métodos IRAM 12 563 y ASTM C-417-58, modificando algunas de las condiciones especificadas; se consideraron como variables el espesor de la probeta de ensayo, el tiempo de calentamiento a la temperatura de régimen y el tratamiento térmico previo de la probeta de ensayo• Las muestras utilizadas son productos nacionales proporcionados por los usuarios. Se pretende así establecer las bases de futuras especificaciones relativas a las características de conductividad térmica y de módulo de rotura a la flexión, valores éstos no fijados en la actualidad.Several experiences were done following the IRAM 12563 and ASTM C-417-58 methods with some modifications in the specified conditions. The thickness of specimens, the heating time to the work temperature and the previous thermic treatment were considered as variables. The samples studied are argentine products. In this way we try to outline the basis for future specifications in connection with the characteristics of thermical conductivity and point of breakage in flexion. These values have not been established yet

Conformal metrics and true "gradient flows" for curves

We wish to endow the manifold M of smooth curves in Rn with a Riemannian metric that allows us to treat continuous morphs (homotopies) between two curves c0 and c1 as trajectories with computable lengths which are independent of the parameterization or representation of the two curves (and the curves making up the morph between them). We may then define the distance between the two curves using the trajectory of minimal length (geodesic) between them, assuming such a minimizing trajectory exists. At first we attempt to utilize the metric structure implied rather unanimously by the past twenty years or so of shape optimization literature in computer vision. This metric arises as the unique metric which validates the common references to a wide variety of contour evolution models in the literature as "gradient flows" to various formulated energy functionals. Surprisingly, this implied metric yields a pathological and useless notion of distance between curves. In this paper, we show how this metric can be minimally modified using conformal factors that depend upon a curve's total arclength. A nice property of these new conformal metrics is that all active contour models that have been called "gradient flows" in the past will constitute true gradient flows with respect to these new metrics under specific time reparameterizations

Separation functions and mild topologies

Given M and N Hausdorff topological spaces, we study topologies on the space C-0 (M; N) of continuous maps f : M? N. We review two classical topologies, the "strong" and the "weak" topology. We propose a definition of "mild topology" that is coarser than the "strong" and finer than the "weak" topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M? N, and affine actions when N = R-n. To define the "mild topology" we use "separation functions;" these "separation functions" are somewhat similar to the usual "distance function d(x, y)" in metric spaces (M, d), but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define "set separation functions" that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C-0(M; N). Other examples are the Sorgenfrey line and the topology of topological manifolds

Hybrid QM/classical models: Methodological advances and new applications

Hybrid methods that combine quantum mechanical descriptions with classical models are very popular in molecular modeling. Such a large diffusion reflects their effectiveness, which over the years has allowed the quantum mechanical description to extend its boundaries to systems of increasing size and to processes of increasing complexity. Despite this success, research in this field is still very active and a number of advances have been made recently, further extending the range of their applications. In this review, we describe such advances and discuss how hybrid methods may continue to improve in the future. The various formulations proposed so far are presented here in a coherent way to underline their common methodological aspects. At the same time, the specificities of the different classical models and of their coupling with the quantum mechanical domain are highlighted and discussed, with special attention to the computational and numerical aspects

Delocalized excitons in natural light harvesting complexes

Natural organisms such as photosynthetic bacteria, algae, and plants employ complex molecular machinery to convert solar energy into biochemical fuel. An important common feature shared by most of these photosynthetic organisms is that they capture photons in the form of excitons typically delocalized over a few to tens of pigment molecules embedded in protein environments of light harvesting complexes (LHCs). Delocalized excitons created in such LHCs remain well protected despite being swayed by environmental fluctuations, and are delivered successfully to their destinations over hundred nanometer length scale distances in about hundred picosecond time scales. Decades of experimental and theoretical investigation have produced a large body of information offering insights into major structural, energetic, and dynamical features contributing to LHCs' extraordinary capability to harness photons using delocalized excitons. The objective of this review is (i) to provide a comprehensive account of major theoretical, computational, and spectroscopic advances that have contributed to this body of knowledge, and (ii) to clarify the issues concerning the role of delocalized excitons in achieving efficient energy transport mechanisms. The focus of this review is on three representative systems, Fenna-Matthews-Olson complex of green sulfur bacteria, light harvesting 2 complex of purple bacteria, and phycobiliproteins of cryptophyte algae. Although we offer more in-depth and detailed description of theoretical and computational aspects, major experimental results and their implications are also assessed in the context of achieving excellent light harvesting functionality. Future theoretical and experimental challenges to be addressed in gaining better understanding and utilization of delocalized excitons are also discussed.Comment: 53 pages, 15 figure

The key to the yellow-to-cyan tuning in the green fluorescent protein family is polarisation

Computational approaches have to date failed to fully capture the large (about 0.4 eV) excitation energy tuning displayed by the nearly identical anionic chromophore in different green fluorescent protein (GFP) variants. Here, we present a thorough comparative study of a set of proteins in this sub-family, including the most red- (phiYFP) and blue-shifted (mTFP0.7) ones. We employ a classical polarisable embedding through induced dipoles and combine it with time-dependent density functional theory and multireference perturbation theory in order to capture both state-specific induction contributions and the coupling of the polarisation of the protein to the chromophore transition density. The obtained results show that only upon inclusion of both these two effects generated by the mutual polarisation between the chromophore and the protein can the full spectral tuning be replicated. We finally discuss how this mutual polarisation affects the correlation between excitation energies, dipole moment variation, and molecular electrostatic field