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Free, publiclyaccessible full text available April 1, 2025

The terminal alkyne C≡C stretch has a large Raman scattering cross section in the “silent” region for biomolecules. This has led to many Raman tag and probe studies using this moiety to study biomolecular systems. A computational investigation of these systems is vital to aid in the interpretation of these results. In this work, we develop a method for computing terminal alkyne vibrational frequencies and isotropic transition polarizabilities that can easily and accurately be applied to any terminal alkyne molecule. We apply the discrete variable representation method to a localized version of the C≡C stretch normal mode. The errors of (1) vibrational localization to the terminal alkyne moiety, (2) anharmonic normal mode isolation, and (3) discretization of the Born–Oppenheimer potential energy surface are quantified and found to be generally small and cancel each other. This results in a method with low error compared to other anharmonic vibrational methods like secondorder vibrational perturbation theory and to experiments. Several density functionals are tested using the method, and TPSSD3, an inexpensive nonempirical density functional with dispersion corrections, is found to perform surprisingly well. Diffuse basis functions are found to be important for the accuracy of computed frequencies. Finally, the computation of vibrational properties like isotropic transition polarizabilities and the universality of the localized normal mode for terminal alkynes are demonstrated.
Free, publiclyaccessible full text available February 21, 2025 
Abstract An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heatstructure interaction partial differential equation (PDE) models, with structure subject to high KelvinVoigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevant
fluid structure models, with wave or platestructures, possibly without KelvinVoigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDEenergy estimates dictate how to define the interface/boundary feedback control in each of the three cases. 
We consider the linear third order (in time) PDE known as the SMGTJequation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control
. Optimal interior and boundary regularity results were given in [\begin{document}$ g $\end{document} 1 ], after [41 ], when , which, moreover, in the canonical case\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document} , were expressed by the wellknown explicit representation formulae of the wave equation in terms of cosine/sine operators [\begin{document}$ \gamma = 0 $\end{document} 19 ], [17 ], [24 ,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether or\begin{document}$ \gamma = 0 $\end{document} , since\begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator basedexplicit representation formulae to provide optimal interior and boundary regularity results with\begin{document}$ \gamma \neq 0 $\end{document} "smoother" than\begin{document}$ g $\end{document} , qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [\begin{document}$ L^2(\Sigma) $\end{document} 17 ]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22 ], [23 ], [37 ] for control smoother than , and [\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document} 44 ] for control less regular in space than . In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [\begin{document}$ L^2(\Gamma) $\end{document} 42 ], [24 ,Section 9.8.2]. 
We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.