Quantum alchemy beyond singlets: Bonding in diatomic molecules with hydrogen

Bonding energies are key for the relative stability of molecules in chemical space. Therefore methods employed to search for relevant molecules in chemical space need to capture the bonding behavior for a wide range of molecules, including radicals. In this work, we investigate the ability of quantum alchemy to do so for exploring hypothetical chemical compounds, here diatomic molecules involving hydrogen with various electronic structures. We evaluate equilibrium bond lengths, ionization ener- gies, and electron affinities of these fundamental systems. We compare and contrast how well manual quantum alchemy calculations, i.e. quantum mechanical calculations in which the nuclear charge is altered, and quantum alchemy approximations using a Taylor series expansion can predict these molecular properties. We also investigate the extent of error cancellation of these approaches in terms of ionization energies and electron affinities when using thermodynamic cycles. Our results suggest that the accuracy of Taylor series expansions are greatly improved by error cancellation in thermodynamic cycles, and errors also appear to be generally system-dependent. Taken together, this work provides insights into how quantum alchemy predictions us- ing a Taylor series expansion may be applied to future studies of non-singlet systems as well as which challenges remain open for these cases.

Adaptive Robust Motion Control of Quadrotor Systems Using Artificial Neural Networks and Particle Swarm Optimization

Most of the mechanical dynamic systems are subjected to parametric uncertainty, unmodeled dynamics, and undesired external vibrating disturbances while are motion controlled. In this regard, new adaptive and robust, advanced control theories have been developed to efficiently regulate the motion trajectories of these dynamic systems while dealing with several kinds of variable disturbances. In this work, a novel adaptive robust neural control design approach for efficient motion trajectory tracking control tasks for a considerably disturbed non-linear under-actuated quadrotor system is introduced. Self-adaptive disturbance signal modeling based on Taylor-series expansions to handle dynamic uncertainty is adopted. Dynamic compensators of planned motion tracking errors are then used for designing a baseline controller with adaptive capabilities provided by three layers B-spline artificial neural networks (Bs-ANN). In the presented adaptive robust control scheme, measurements of position signals are only required. Moreover, real-time accurate estimation of time-varying disturbances and time derivatives of error signals are unnecessary. Integral reconstructors of velocity error signals are properly integrated in the output error signal feedback control scheme. In addition, the appropriate combination of several mathematical tools, such as particle swarm optimization (PSO), Bézier polynomials, artificial neural networks, and Taylor-series expansions, are advantageously exploited in the proposed control design perspective. In this fashion, the present contribution introduces a new adaptive desired motion tracking control solution based on B-spline neural networks, along with dynamic tracking error compensators for quadrotor non-linear systems. Several numeric experiments were performed to assess and highlight the effectiveness of the adaptive robust motion tracking control for a quadrotor unmanned aerial vehicle while subjected to undesired vibrating disturbances. Experiments include important scenarios that commonly face the quadrotors as path and trajectory tracking, take-off and landing, variations of the quadrotor nominal mass and basic navigation. Obtained results evidence a satisfactory quadrotor motion control while acceptable attenuation levels of vibrating disturbances are exhibited.

Modularity of the displacement coefficients and complete plate theories in the framework of the consistent-approximation approach

Starting from the three-dimensional theory of linear elasticity, we arrive at the exact plate problem by the use of Taylor series expansions. Applying the consistent-approximation approach to this problem leads to hierarchic generic plate theories. Mathematically, these plate theories are systems of partial-differential equations (PDEs), which contain the coefficients of the series expansions of the displacements (displacement coefficients) as variables. With the pseudo-reduction method, the PDE systems can be reduced to one main PDE, which is entirely written in the main variable, and several reduction PDEs, each written in the main variable and several non-main variables. So, after solving the main PDE, the reduction PDEs can be solved by insertion of the main variable. As a great disadvantage of the generic plate theories, there are fewer reduction PDEs than non-main variables so that not all of the latter can be determined independently. Within this paper, a modular structure of the displacement coefficients is found and proved. Based on it, we define so-called complete plate theories which enable us to determine all non-main variables independently. Also, a scheme to assemble Nth-order complete plate theories with equations from the generic plate theories is found. As it turns out, the governing PDEs from the complete plate theories fulfill the local boundary conditions and the local form of the equilibrium equations a priori. Furthermore, these results are compared with those of the classical theories and recently published papers on the consistent-approximation approach.

A THEORETICAL STUDY OF OSCILLATIONS OF TWO IMMISCIBLE FLUIDS IN A LIMITED TANK

In the paper, the nonlinear oscillations of a two-layer fluid that completely fills a limited tank are theoretically studied. To determine any smooth function on the deflected interface, the Taylor series expansions are considered using the values of the function and its normal derivatives on the undisturbed interface of the fluids. Using two fundamental asymmetric harmonics, which are generated in two mutually perpendicular planes, the differential equations of nonlinear oscillations of the two-layer fluid interface are investigated. As a result, the frequency-response characteristics are presented and the instability regions of the forced oscillations of the two-layer fluid in the cylindrical tank are plotted, as well as the parametric resonance regions for different densities of the upper and lower fluids. The Bubnov-Galerkin method is used to plot instability regions for the approximate solution to nonlinear differential equations. At the final stage of the work, the nonlinear effects resulting from the interaction of fluids with a rigid tank that executes harmonic oscillations at the interface of the fluids are theoretically studied.

Computational Analysis of Generalized Zeta Functions by Using Difference Equations

In this article, author performs computational analysis for the generalized zeta functions by using computational software Mathematica. To achieve the purpose recently obtained difference equations are used. These difference equations have a computational power to compute these functions accurately while they can not be computed by using their known integral represenations. Several authors investigated such functions and their analytic properties, but no work has been reported to study the graphical representations and zeors of these functions. Author performs numerical computations to evaluate these functions for different values of the involved parameters. Taylor series expansions are also presented in this research.

Hadamard parametrix of the Feynman Green’s function of a five-dimensional charged scalar field

The Hadamard parametrix is a representation of the short-distance singularity structure of the Feynman Green’s function for a quantum field on a curved spacetime background. Subtracting these divergent terms regularizes the Feynman Green’s function and enables the computation of renormalized expectation values of observables. We study the Hadamard parametrix for a charged, massive, complex scalar field in five spacetime dimensions. Even in Minkowski spacetime, it is not possible to write the Feynman Green’s function for a charged scalar field exactly in closed form. We, therefore, present covariant Taylor series expansions for the biscalars arising in the Hadamard parametrix. On a general spacetime background, we explicitly state the expansion coefficients up to the order required for the computation of the renormalized scalar field current. These coefficients become increasingly lengthy as the order of the expansion increases, so we give the higher-order terms required for the calculation of the renormalized stress-energy tensor in Minkowski spacetime only.

Fault estimation for a class of nonlinear time-variant systems through a Krein space–based approach

This paper studies the [Formula: see text] fault estimation problem for a class of discrete-time nonlinear systems subject to time-variant coefficient matrices, online available input, and exogenous disturbances. By assuming that the concerned nonlinearity is continuously differentiable and by using Taylor series expansions, the dynamic system is transferred as a linear time-variant system with modeling uncertainties. A non-conservative but nominal system and its corresponding [Formula: see text] indefinite quadratic performance function are, respectively, given in place of the transferred uncertain system and the conventional performance metric, such that the estimation problem is converted as a two-stage optimization issue. By introducing an auxiliary model in Krein space, the so-called orthogonal projection technique is utilized to search an appropriate choice serving as the estimation of the fault signal. A necessary and sufficient condition on the existence of the fault estimator is given, and a recursive algorithm for computing the gain matrix of the estimator is proposed. The addressed method is applied to an indoor robot localization system to show its effectiveness.

Comfort for the Computationally Crippled

The difference between engineering and science, and all other human activity, is the fact that engineers and scientists make quantitative predictions about measurable outcomes and can specify their uncertainty in such predictions. Because those predictions are quantitative, they must employ mathematics. This chapter is intended as review of some of the more useful mathematical concepts, strategies, and techniques that are employed in the description of vibrational and acoustical systems and in the calculation of their behavior. Topics in this review include techniques such as Taylor series expansions, integration by parts, and logarithmic differentiation. Equilibrium and stability considerations lead to relations between potential energies and forces. The concept of linearity leads to superposition and Fourier analysis. Complex numbers and phasors are introduced along with the techniques for their algebraic manipulation. The discussion of physical units is extended to include their use for predicting functional dependencies of resonance frequencies, quality factors, propagation speeds, flow noise, and other system behaviors using similitude and the Buckingham Π-theorem to form dimensionless variables. Linearized least-squares fitting is introduced as a method for extraction of experimental parameters and their uncertainties and error propagation is presented to allow those uncertainties to be combined.

Quantum mechanics at high redshift – Modelling Damped Lyman–α absorption systems

For around 100 years, hydrogen spectral modelling has been based on Voigt profile fitting. The semi-classical Voigt profile is based on a 2-level atom approximation. Whilst the Voigt profile is excellent for many circumstances, the accuracy is insufficient for very high column density damped Lyman-α absorption systems. We have adapted the quantum-mechanical Kramers-Heisenberg model to include thermal broadening, producing a new profile, the KHT profile. Interactions involving multiple discrete atomic levels and continuum terms, not accounted for in the Voigt model, generate asymmetries in the Lyman line wings. If not modelled, this can lead to significant systematics in parameter estimation when modelling real data. There are important ramifications in particular for measurements of the primordial deuterium abundance. However, the KHT model is complicated. We therefore present a simplified formulation based on Taylor series expansions and look-up tables, quantifying the impact of the approximations made. The KHT profile has been implemented within the widely-used VPFIT code.

Estimation for Two-Dimensional Incoherently Distributed Source in Double L-Shape Arrays

Distributed sources can be regarded as an assembly of point sources within a spatial distribution. In this paper, we explore the estimation of the two-dimensional incoherently distributed sources using double L-shape arrays. The rotational invariance properties of the nominal elevation and nominal elevation are firstly obtained by taking first-order Taylor series expansions with regard to the generalized steering vectors of two pairs of parallel subarrays. The rotation operators can be solved based on signal subspace. Then the nominal elevation and nominal elevation can be obtained from parameters matching method. Estimation of direction of arrival can be used in multi-source scenario and needn't peak-finding search. Lastly the angular spreads can be solved through two-dimensional Capon search based on nominal angles. The simulation experiments show that the proposed method has good performance on the estimation of two-dimensional incoherently distributed sources. Investigating different experimental conditions, sources with different angular spreads, simulations are conducted to validate better estimation accuracy of the proposed method.