This paper concerns the representation of angular momentum operators in the Born–Oppenheimer theory of polyatomic molecules and the various forms of the associated conservation laws. Topics addressed include the question of whether these conservation laws are exactly equivalent or only to some order of the Born–Oppenheimer parameter κ = ( m/ M) 1/4 and what the correlation is between angular momentum quantum numbers in the various representations. These questions are addressed in both problems involving a single potential energy surface and those with multiple, strongly coupled surfaces and in both the electrostatic model and those for which fine structure and electron spin are important. The analysis leads to an examination of the transformation laws under rotations of the electronic Hamiltonian; of the basis states, both adiabatic and diabatic, along with their phase conventions; of the potential energy matrix; and of the derivative couplings. These transformation laws are placed in the geometrical context of the structures in the nuclear configuration space that are induced by rotations, which include the rotational orbits or fibers, the surfaces upon which the orientation of the molecule changes but not its shape, and the section, an initial value surface that cuts transversally through the fibers. Finally, it is suggested that the usual Born–Oppenheimer approximation can be replaced by a dressing transformation, that is, a sequence of unitary transformations that block-diagonalize the Hamiltonian. When the dressing transformation is carried out, we find that the angular momentum operator does not change. This is a part of a system of exact equivalences among various representations of angular momentum operators in Born–Oppenheimer theory. Our analysis accommodates large-amplitude motions and is not dependent on small-amplitude expansions about an equilibrium position. Our analysis applies to noncollinear configurations of a polyatomic molecule; this covers all but a subset of measure zero (the collinear configurations) in the nuclear configuration space.
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Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential
This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.
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- Award ID(s):
- 1945224
- NSF-PAR ID:
- 10316563
- Date Published:
- Journal Name:
- Quarterly of Applied Mathematics
- Volume:
- 80
- Issue:
- 1
- ISSN:
- 0033-569X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The past six years have seen the rapid growth of studies of CRISPR/Cas9 in plant genome editing, a method that enormously facilitates both basic research and practical applications. Most studies have focused on genetic model species, but plant species that are not genetic models may also be economically important or biologically significant, or both. However, developing the CRISPR/Cas9 system in a nongenetic model is challenging. Here, we summarize CRISPR/Cas9 applications in 45 plant genera across 24 families and provide a reference for practical application of CRISPR in nongenetic model plant systems. Suggestions for selecting plant species and target genes are given for proof‐of‐principle CRISPR studies, and the processes of vector construction are reviewed. We recommend using transient assays to identify a desired CRISPR/Cas9 system in a nongenetic model. We then review methods of plant transformation and describe approaches, using regenerated transgenic plants, for evaluating CRISPR editing results. Lastly, potential future applications of CRISPR in nongenetic model plant species are discussed. This review provides a road map for developing CRISPR in nongenetic models, an application that holds enormous potential in plant biology.
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Abstract Despite the well‐recognized initial value nature of the subseasonal forecasts, the role of subsurface ocean initialization in subseasonal forecasts remains underexplored. Using observing system experiments, this study investigates the impact of ocean in situ data assimilation on the propagation of Madden–Julian Oscillation (MJO) events across the Maritime Continent in the European Centre for Medium‐Range Weather Forecasts (ECMWF) subseasonal forecast system. Two sets of twin experiments are analyzed, which only differ on the use or not of in situ ocean observations in the initial conditions. Besides using the Real‐time Multivariate MJO Index (RMMI) to evaluate the forecast performance, we also develop a new MJO tracking method based on outgoing longwave radiation anomalies (OLRa) for forecast evaluation. We find that the ocean initialization with in situ data assimilation, though having an impact on the forecasted ocean mean state, does not improve the relatively low MJO forecast skill across the Maritime Continent. Moist static energy budget analysis further suggests that a significant underestimation in the meridional moisture advection in the model forecast may hinder the potential role played by the ocean state differences associated with data assimilation. Bias of the intraseasonal meridional winds in the model is a more important factor for such underestimation than the mean state moisture biases. This finding suggests that atmospheric model biases dominate the forecast error growth, and the atmospheric circulation bias is one of the major sources of the MJO prediction error and should be a target for improving the ECMWF subseasonal forecast model.
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The effect of initial microstructure and its evolution across the α→ω phase transformation in commercially pure Zr under hydrostatic compression has been studied using in situ x-ray diffraction measurements. Two samples were studied: one is plastically pre-deformed Zr with saturated hardness and the other is annealed. Phase transformation α→ω initiates at lower pressure for pre-deformed sample, suggesting pre-straining promotes nucleation by producing more defects with stronger stress concentrators. With transformation progress, the promoting effect on nucleation reduces while that on growth is suppressed by producing more obstacles for interface propagation. The crystal domain size reduces and microstrain and dislocation density increase during loading for both α and ω phases in their single-phase regions. For α phase, domain sizes are much smaller for prestrained Zr, while microstrain and dislocation densities are much higher. On the other hand, they do not differ much in ω Zr for both prestrained and annealed samples, implying that microstructure is not inherited during phase transformation. The significant effect of pressure on the microstructural parameters (domain size, microstrain, and dislocation density) demonstrates that their postmortem evaluation does not represent the true conditions during loading. A simple model for the initiation of the phase transformation involving microstrain is suggested, and a possible model for the growth is outlined. The obtained results suggest an extended experimental basis is required for better predictive models for the pressure-induced and combined pressure- and strain-induced phase transformations.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/qam/1600
