Regularity estimates on harmonic eigenmaps with arbitrary number of coordinates

We continue the study of null-vector equations in relation with partition functions of (systems of) Schramm-Loewner Evolutions (SLEs) by considering the question of fusion. Starting from $n$ commuting SLEs seeded at distinct points, the partition function satisfies $n$ null-vector equations (at level 2). We show how to obtain higher level null-vector equations by coalescing the seeds one by one. As an example, we extend Schramm's formula (for the position of a marked bulk point relatively to a chordal SLE trace) to an arbitrary number of SLE strands. The argument combines input from representation theory - the study of Verma modules for the Virasoro algebra - with regularity estimates, themselves based on hypoellipticity and stochastic flow arguments.

The incorporation of the geometric phase in single-state adiabatic dynamics near a conical intersection (CI) seam has so far been restricted to molecular systems with high symmetry or simple model Hamiltonians. This is due to the fact that the ab initio determined derivative coupling (DC) in a multi-dimensional space is not curl-free, thus making its line integral path dependent. In a recent work [C. L. Malbon et al., J. Chem. Phys. 145, 234111 (2016)], we proposed a new and general approach based on an ab initio determined diabatic representation consisting of only two electronic states, in which the DC is completely removable, so that its line integral is path independent in the simply connected domains that exclude the CI seam. Then with the CIs included, the line integral of the single-valued DC can be used to construct the complex geometry-dependent phase needed to exactly eliminate the double-valued character of the real-valued adiabatic electronic wavefunction. This geometry-dependent phase gives rise to a vector potential which, when included in the adiabatic representation, rigorously accounts for the geometric phase in a system with an arbitrary locus of the CI seam and an arbitrary number of internal coordinates. In this work, we demonstrate this approach in a three-dimensional treatment of the tunneling facilitated dissociation of the S1 state of phenol, which is affected by a Cs symmetry allowed but otherwise accidental seam of CI. Here, since the space is three-dimensional rather than two-dimensional, the seam is a curve rather than a point. The nodal structure of the ground state vibronic wavefunction is shown to map out the seam of CI.

The generalized Haag theorem was proven in SO(1, k) invariant quantum field theory. Apart from the k + 1 variables, an arbitrary number of additional coordinates, including noncommutative ones, can occur in the theory. In SO(1, k) invariant theory new corollaries of the generalized Haag theorem are obtained. It has been proven that the equality of four-point Wightman functions in the two theories leads to the equality of elastic scattering amplitudes and thus to the equality of the total cross sections in these theories. It was also shown that at k > 3 the equality of (k + 1) point Wightman functions in the two theories leads to the equality of the scattering amplitudes of some inelastic processes. In the SO(1, 1) invariant theory it was proven that if in one of the theories under consideration the S-matrix is equal to unity, then in another theory the S-matrix equals unity as well.

There exists a generalization of Boltzmann's H-function that allows for nonuniformly populated stationary states, which may exist far from thermodynamic equilibrium. Here we describe a method for obtaining a generalized or collective diffusion coefficient D directly from this H-function, the only constraints being that the relaxation process is Markov (short memory), continuous in the reaction coordinate, and local in the sense of a flux/force relationship. As an application of this H-function method, we simulate the self-consistent extraction of D via Langevin/Fokker-Planck (L/FP) dynamics on various potential energy landscapes. We observe that the initial epoch of relaxation, which is far removed from the stationary state, provides the most reliable estimates of D. The construction of an H-function that guarantees conformity with the second law of thermodynamics has been generalized to allow for diffusion coefficients that may depend on both the reaction coordinate and time, and the extension to an arbitrary number of reaction coordinates is straightforward. For this multidimensional case, the diffusion tensor must be positive definite in the sense that its eigenvalues must be real and positive. To illustrate the behavior of the proposed collective diffusion coefficient, we simulate the H-function for a variety of Langevin systems. In particular, the impacts on H and D of landscape shape, sample size, selection of an initial distribution, finite dynamic observation range, stochastic correlations, and short/long-term memory effects are examined.

A mathematical model of a flexible-link manipulator is studied in this chapter within the framework of the Timoshenko beam theory. In contrast to the majority of publications in this area, we consider here the model with two rigid bodies and the beam with non-collocated sensors and actuators. For the mathematical model described by coupled ordinary and partial differential equations, we construct a family of Galerkin’s approximations with an arbitrary number of modal coordinates. Asymptotic properties of the eigenvalues of the associated spectral problem are investigated for the case of a homogeneous beam. We propose a state feedback law and an observer-based stabilization scheme for Galerkin’s systems.

The paper is devoted to the observability study of a dynamic system, which describes the vibrations of an elastic beam with an attached rigid body and distributed control actions. The mathematical model is derived using Hamilton's principle in the form of the Euler-Bernoulli beam equation with hinged boundary conditions and interface condition at the point of attachment of the rigid body. It is assumed that the sensors distributed along the beam provide output information about the deformation in neighborhoods of the specified points of the beam. Based on the variational form of the equations of motion, the spectral problem for defining the eigenfrequencies and eigenfunctions of the beam oscillations is obtained. Some properties of the eigenvalues and eigenfunctions of the spectral problem are investigated. Finite-dimensional approximations of the dynamic equations are constructed in the linear manifold spanned by the system of eigenfunctions. For these Galerkin approximations, observability conditions for the control system with incomplete information about the state are derived. An algorithm for observer design with an arbitrary number of modal coordinates is proposed for the differential equation on a finite-dimensional manifold. Based on a quadratic Lyapunov function with respect to the coordinates of the finite-dimensional state vector, the exponential convergence of the observer dynamics is proved. The proposed method of constructing a dynamic observer makes it possible to estimate the full system state by the output signals characterizing the motion of particular point only. Numerical simulations illustrate the exponential decay of the norm of solutions of the system of ordinary differential equations that describes the observation error.

Mode mixing via resonance Raman excitation profiles

A new spectral-based multi-substructure theory is formulated to compute the frequency responses of mechanical systems that are subdivided into multiple inter-connected substructures. The proposed approach employs the free substructure frequency response functions at the coupling, response and excitation coordinates of interest to construct the complete system model using a single efficient coupling step. Even though this proposed approach is conceptually similar to the conventional transfer path analysis, it is more extensive because of the capability to analyze structural systems with arbitrary numbers of substructures and coupling coordinates. Hence, the proposed methodology can be applied to treat complex multi-substructure mechanical structures commonly found in automotive and aerospace systems. In the present study, several lumped parameters mass-spring-damper systems are analyzed to validate the proposed theory. The comparison results show excellent agreement between the multi-substructure predictions and the single complete system calculations.

An irreversible thermodynamics formulation for a fluid with an arbitrary number of internal coordinates leads to an inhomogeneous wave equation for acoustic pressure with source term, −(β/cp) ∂q/∂t − (K/cp)∂2q/∂t2. Here β is coefficient of thermal expansion, q is heat added per unit time and volume, and K is a constant of the order of 10−11 s/°C which is related to the relaxation time matrix. The theory explains the tripolar pressure pulses observed by Hunter et al. [J. Acoust. Soc. Am. 69, 1563–1567 (1981)] following irradiation of 4 °C water (for which β = 0) by short laser pulses. A precise evaluation of K follows from analysis of reported waveforms. If water has only a single relaxation process, the magnitude of K is at variance, however, with the Hall structural relaxation theory of bulk viscosity, for which the estimated relaxation time is of the order of 10−12 s. The conflict is resolved if water has a second relaxation process that has negligible influence on absorption and wave speed, but which substantially influences β. The relaxation time for this newly inferred process may be as long as 10−6 s. [Work supported by ONR.]

The generalized traveling wave method (GTWM) is developed for the nonlinear Schrödinger equation (NLSE) with general perturbations in order to obtain the equations of motion for an arbitrary number of collective coordinates. Regardless of the particular ansatz that is used, it is shown that this alternative approach is equivalent to the Lagrangian formalism, but has the advantage that only the Hamiltonian of the unperturbed system is required, instead of the Lagrangian for the perturbed system. As an explicit example, we take 4 collective coordinates, namely the position, velocity, amplitude and phase of the soliton, and show that the GTWM yields the same equations of motion as the perturbation theory based on the Inverse Scattering Transform and as the time variation of the norm, first moment of the norm, momentum, and energy for the perturbed NLSE.

A stochastic mathematical model of the control system for an arbitrary number of coordinates of the object under study and control parameters is obtained. The developed mathematical model allows one to fully take into account the features of random sequences of changes of coordinates and control parameters, as well as to fully use all known a posteriori and a priori data about the control object. To obtain the model, the apparatus of nonlinear vector canonical expansions is used. The work presents diagrams that reflect the specifics of determining the parameters of a multiparameter stochastic control model and the regularities of its functioning. The obtained mathematical model has wide possibilities of practical application for solving problems of controlling objects of various nature in conditions of uncertainty.

We propose a random coordinate descent algorithm for optimizing a non-convex objective function subject to one linear constraint and simple bounds on the variables. Although it is common use to update only two random coordinates simultaneously in each iteration of a coordinate descent algorithm, our algorithm allows updating arbitrary number of coordinates. We provide a proof of convergence of the algorithm. The convergence rate of the algorithm improves when we update more coordinates per iteration. Numerical experiments on large scale instances of different optimization problems show the benefit of updating many coordinates simultaneously.

A rigorous expression is obtained which relates time-correlation function of an arbitrary number of coordinates to the correlation of the time derivatives of the original function. It is argued that the formal result can be approximated in several ways: by assuming that the time-decay of the molecular velocities is fast compared to coordinate changes; by assuming that velocities of different degrees of freedom are uncorrelated; and by viewing the approximate expressions as leading terms in a series which can be closed using a cumulant formalism. The general arguments are applied to some specific cases including: the anisotropic rotation of a symmetric top molecule; the intermediate scattering factor for an atom in a molecule; and the combined translation-rotation correlation function that appears in the theory for quadrupole-induced infrared absorption spectrum.

We recently introduced an umbrella sampling method for obtaining nonequilibrium steady-state probability distributions projected onto an arbitrary number of coordinates that characterize a system (order parameters) [A. Warmflash, P. Bhimalapuram, and A. R. Dinner, J. Chem. Phys. 127, 154112 (2007)]. Here, we show how our algorithm can be combined with the image update procedure from the finite-temperature string method for reversible processes [E. Vanden-Eijnden and M. Venturoli, "Revisiting the finite temperature string method for calculation of reaction tubes and free energies," J. Chem. Phys. (in press)] to enable restricted sampling of a nonequilibrium steady state in the vicinity of a path in a many-dimensional space of order parameters. For the study of transitions between stable states, the adapted algorithm results in improved scaling with the number of order parameters and the ability to progressively refine the regions of enforced sampling. We demonstrate the algorithm by applying it to a two-dimensional model of driven Brownian motion and a coarse-grained (Ising) model for nucleation under shear. It is found that the choice of order parameters can significantly affect the convergence of the simulation; local magnetization variables other than those used previously for sampling transition paths in Ising systems are needed to ensure that the reactive flux is primarily contained within a tube in the space of order parameters. The relation of this method to other algorithms that sample the statistics of path ensembles is discussed.

Vibrations and waves in laminated orthotropic circular cylinders