Coordinate-dependent Coefficients Research Articles

Machine Learning Seams of Conical Intersection: A Characteristic Polynomial Approach.

The machine learning of potential energy surfaces (PESs) has undergone rapid progress in recent years. The vast majority of this work, however, has been focused on the learning of ground state PESs. To reliably extend machine learning protocols to excited state PESs, the occurrence of seams of conical intersections between adiabatic electronic states must be correctly accounted for. This introduces a serious problem, for at such points, the adiabatic potentials are not differentiable to any order, complicating the application of standard machine learning methods. We show that this issue may be overcome by instead learning the coordinate-dependent coefficients of the characteristic polynomial of a simple decomposition of the potential matrix. We demonstrate that, through this approach, quantitatively accurate machine learning models of seams of conical intersection may be constructed.

Coordinate-Dependent Drift-Diffusion Reveals the Kinetic Intermediate Traps of Top7-Based Proteins.

The computer-designed Top7 served as a scaffold to produce immunoreactive proteins by grafting of the 2F5 HIV-1 antibody epitope (Top7-2F5) followed by biotinylation (Top7-2F5-biotin). The resulting nonimmunoglobulin affinity proteins were effective in inducing and detecting the HIV-1 antibody. However, the grafted Top7-2F5 design led to protein aggregation, as opposed to the soluble biotinylated Top7-2F5-biotin. The structure-based model predicted that the thermodynamic cooperativity of Top7 increases after grafting and biotin-labeling, reducing their intermediate state populations. In this work, the folding kinetic traps that might contribute to the aggregation propensity are investigated by the diffusion theory. Since the engineered proteins have similar sequence and structural homology, they served as protein models to study the kinetic intermediate traps that were uncovered by characterizing the position-dependent drift-velocity (v(Q)) and the diffusion (D(Q)) coefficients. These coordinate-dependent coefficients were taken into account to obtain the folding and transition path times over the free energy transition states containing the intermediate kinetic traps. This analysis may be useful to predict the aggregated kinetic traps of scaffold-epitope proteins that might compose novel diagnostic and therapeutic platforms.

Computing vibrational energy levels of CH4 with a Smolyak collocation method.

In this paper, we demonstrate that it is possible to apply collocation to compute vibrational energy levels of a five-atom molecule using an exact kinetic energy operator (with cross terms and coordinate-dependent coefficients). This is made possible by using (1) a pruned basis of products of univariate functions; (2) a Smolyak grid made from nested sequences of grids for each coordinate; (3) a collocation method that obviates the need to solve a generalized eigenvalue problem; (4) an efficient sequential transformation between the (nondirect product) grid and the (nondirect product) basis representations; and (5) hierarchical univariate functions that make it possible to avoid storing large intermediate vectors. The accuracy of the method is confirmed by computing 500 vibrational energy levels of methane.

Oscillations of a string on an elastic foundation with space and time-varying rigidity

The dynamics of a string on an elastic foundation with time- and coordinate-dependent coefficients have been studied. Asymptotic solutions have been constructed for the following cases: for an arbitrary value of the elastic foundation coefficient at small and large time values, and for small and large coefficients of the elastic foundation at arbitrary times. Also a special case originated from an ageing process has been studied. The ageing process is described by an expression approximating some well-known experimental data. The existence of localized modes along the x coordinate is shown. The existence of these localized modes can lead to a spatial resonance phenomenon under certain conditions. For the case of an arbitrary elastic foundation coefficient value at small and at large times, the spatial resonance phenomenon is observed at small, special frequencies. This effect depends also on a special phase and mode number. For large mode numbers, this special resonance seems to be not possible.

Casimir energy–momentum tensor for a brane in de Sitter spacetime

Vacuum expectation values of the energy–momentum tensor for a conformally coupled scalar field are investigated in de Sitter (dS) spacetime in presence of a curved brane on which the field obeys the Robin boundary condition with coordinate dependent coefficients. To generate the corresponding vacuum densities we use the conformal relation between dS and Rindler spacetimes and the results previously obtained by one of the authors for the Rindler counterpart. The resulting energy–momentum tensor is non-diagonal and induces anisotropic vacuum stresses. The asymptotic behaviour of this tensor is investigated near the dS horizon and the boundary.

Hidden integrable hierarchies of AKNS type

A method for constructing integrable hierarchies by restricting AKNS flows on manifolds of finite codimension in the space of independent parameters is provided. Two particular types of hierachies are characterized: one is given by nonlinear differential equations with coordinate-dependent coefficients and the other is related to Schrodinger spectral problems with energy-dependent potentials of even degree.

Multidimensional soliton equations in inhomogeneous media

We use the general formalism of the ∂ -problem to derive nonlinear PDEs that are soliton equations with coordinate-dependent coefficients. Examples of these novel equations are a reduction of the Darboux equations, and a NWRI-type system.

Long-Range Behavior of Nuclear Forces as a Manifestation of Supersymmetry in Nature

It is shown that each isospin channel in the long-range approximation to the nucleon-nucleon potential (one-pion-exchange potential) corresponds to a realization of a quantum mechanical supersymmetric Hamiltonian. A functional relation between the coordinate-dependent coefficients of the spin-spin and tensor parts of the interaction predicted by supersymmetry is exactly fulfilled.

Soliton solutions of the non linear Schrödinger equation with an external time-dependent field

A functional transformation between solutions of the one-dimensional nonlinear Schrödinger equation with time- and coordinate-dependent coefficients and solutions of the conventional nonlinear Schrödinger equation (NSE) is constructed. Exact solutions of the NSE with a homogeneous time-dependent external electric field and the NSE with oscillator potential are obtained.

Two-dimensional non-linear evolution equations: The derivation and the transient wave solution

The general theory of two-dimensional evolution equations describing transient wave propagation in non-linear continuous media is presented. The ray method is used and the two-dimensional evolution equations for plane and cylindrical wave-beams are obtained. The transient wave solutions are discussed briefly. A transformation of variables is proposed that permits the transformation of the two-dimensional evolution equation into a one-dimensional evolution equation with coordinate-dependent coefficients. A breakdown time analysis is carried out for this case. The dispersion relations for plane and cylindrical wave-beams are presented. The non-linear dispersion relation is obtained by making use of a series representation.

Solutions of sine-gordon equation with particular coordinate dependent coefficients in n-dimensional space

Solutions of sine-gordon equation with particular coordinate dependent coefficients in n-dimensional space