Über die kreiseleigenschaften des atomkerns

This paper presents a versatile hyper-ellipsoidal basis function for function approximation in a given high dimensional space. This hyper-ellipsoidal basis function can be translated and rotated to cover the data based upon the distribution of data in a given high dimensional space. Based on this function, we propose a one-pass hyper-ellipsoidal learning algorithm for which any new incoming data can be fed for learning without involving the previously learned one. This learning algorithm is used to adjust the parameters of the versatile hyper-ellipsoidal basis function. In addition, we propose the hyper-ellipsoidal basis function (HEBF) neural network that uses the one-pass hyper-ellipsoidal neural learning algorithm. The structure of this neural network is similar to the radial basis function (RBF) neural networks. The hidden neurons in the HEBF neural network can be increased or decreased during learning process. The number of the hidden neurons in the network can be grown based on geometric growth criterion and can be reduced by merging the two hidden neurons into a new hidden neuron based on merging criterion during learning process. The merging process can be done independently without considering the learned data set.KeywordsRadial Basis FunctionHide NeuronRadial Basis Function Neural NetworkHigh Dimensional SpaceInitial WidthThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Approximate wave functions for the hydrogen molecule ground state previously available in the literature are analyzed quantitatively into approximate natural spin orbital functions with particular attention to the corresponding occupation numbers. The analysis demonstrates the very great similarity of all such trial wave functions, and especially the largely molecular orbital nature of the wave function. In addition it shows a close relationship between the molecular orbital and valence bond functions, and the importance for allowing for angular correlation of the electrons by including terms dependent upon the azimuthal coordinate. The analysis particularly demonstrates that approximate natural spin orbital occupation numbers are nearly invariant under a wide variety of choices of basis functions, and therefore are particularly suitable for comparison of different approximate functions and for discussion of their respective properties.

This paper considered the solution of special second order delay differential equations. The method was generated by interpolation and collocation techniques using a combination of power series and exponential function. The approximate basis function is interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve special second order delay differential equations. The basic properties of the methods such as order, error constants, consistency and convergence were also examined. Numerical results are presented to show that the proposed method is suitable for solving second order delay differential equations.

Form invariant stress-strain relationships are presented for anisotropic media. The case of transversely isotropic media is then considered and the five elastic constants needed to describe such a medium are obtained in terms of the elements of the stiffness tensor. The dispersion equation is solved and approximate basis functions are constructed to expand the displacement fields inside such an anisotropic inclusion. The scattering of elastic waves by spherical or nonspherical inclusions is studied as a function of frequency and geometry and compared with the isotropic case.

In this paper, a robust L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> filtering method is proposed for a class of switched linear parameter varying (LPV) systems in which the state-space matrices and the time delay rely on real-time measured parameter. By using multiple delay-dependent Lyapunov-Krasovskii function, the delay-dependent L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> performance criterion for the switched LPV systems is first established. As there exists coupling between Lyapunov function and system parameter matrices, we utilized a slack matrix variable to decouple it. Because of the dependence on parameter, approximate basis function and gridding technique are utilized, and the solution for the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> filters can be transformed into finite dimensional convex optimization problem. Based on the switching logic with the minimum average dwell time (ADT) and parameter linear matrix inequalities (PLMIs) technique, the resulting filters guarantee the filtering error systems to be exponentially stable with a prespecified L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> disturbance attenuation level. A numerical example is presented to verify the effectiveness of the proposed method.

In this work we devise a theoretical and computational method to compute the elastic scattering of electrons from a non-spherical potential, such as in the case of molecules and molecular aggregates. Its main feature is represented by the ability of calculating accurate wave functions for continuum states of polycentric systems via the solution of the Lippmann-Schwinger equation, including both the correlation effects and multi-scattering interference terms, typically neglected in widely used approaches, such as the Mott theory. Within this framework, we calculate the purely elastic scattering matrix elements. As a test case, we apply our scheme to the modelling of electron-water elastic scattering. The Dirac-Hartree-Fock self-consistent field method is used to determine the non-spherical molecular potential projected on a functional space spanned by Gaussian basis set. By adding a number of multi-centric radially-arranged s-type Gaussian functions, whose exponents are system-dependent and optimized to reproduce the properties of the continuum electron wave function in different energy regions, we are able to achieve unprecedented access to the description of the low energy range of the spectrum (0.001 &amp;lt; E &amp;lt; 10 eV) up to keV, finding a good agreement with experimental data and previous theoretical results. To show the potential of our approach, we also compute the total elastic scattering cross section of electrons impinging on clusters of water molecules and zundel cation. Our method can be extended to deal with inelastic scattering events and heavy-charged particles.

Modified Jacobi–Bernstein basis transformation and its application to multi-degree reduction of Bézier curves

A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-to-one correspondence between the continuous piecewise polynomial space of degree k + 1 and the divergencefree vector space of degree k, one can construct high-order two-dimensional exponentially fitted basis functions that are strictly interpolative at a selected node set but are discontinuous on edges in general, spanning nonconforming finite element spaces. First order convergence was proved for the methods constructed from divergence-free Raviart-Thomas space RT 0 0 at two different node sets.

Isogeometric analysis (IGA) is a computational technique that integrates computer-aided design (CAD) with finite element analysis (FEA) by employing the same basis functions for both geometry representation and solution approximation. While IGA offers numerous advantages, such as improved accuracy and efficiency, it also presents several challenges related to geometric modeling. Some of these challenges include accurately representing complex geometries with NURBS (Non-Uniform Rational B-Splines) or other basis functions used in IGA and generating high-quality meshes that conform to the complex geometry represented by NURBS curves/surfaces. This paper introduces an analytical framework to provide a more efficient and theoretically grounded method for generating curvilinear configurations and its analytical solution in IGA, bridging the gap between generated data and its physical representations. This innovative approach is distinguished by integrating the NURBS parameterization in curve generation and providing a corresponding framework to achieve a broader and more accurate explanation of meshes and properties, especially constructing new coordinates and calculating the physical displacements under these conditions. Our model enables the analytical understanding of complex curves from the UIUC airfoil and superformula datasets, demonstrating a deeper dive into simulations. This study signifies a pivotal juncture wherein machine-learning-based complex geometrical formulations are synergistically combined with actual isogeometric analysis.

Parameter optimization of orthonormal basis functions for efficient rational approximations

This paper introduces a tomographic approach for reconstruction of diffusion propagators, P( r ), in a box spline framework. Box splines are chosen as basis functions for high-order approximation of P( r ) from the diffusion signal. Box splines are a generalization of B-splines to multivariate setting that are particularly useful in the context of tomographic reconstruction. The X-Ray or Radon transform of a (tensor-product B-spline or a non-separable) box spline is a box spline - the space of box splines is closed under the Radon transform.We present synthetic and real multi-shell diffusion-weighted MR data experiments that demonstrate the increased accuracy of P( r ) reconstruction as the order of basis functions is increased.

Hair designing is one of crucial components of the hair simulation tasks.The efficiency of hair modeling is very much determined by the interactivity and the ease-to-use the designing tools within an application.This paper presents a unified framework that uses the various key techniques developed for specific tasks in hair simulation and to realize the ultimate goal of 'virtual hairdressing room' that is simple to use but quite effective for generating fast hairstyles.Successful attempts have been made to handle the different challenging issues involved in simulation of hair at interactive rates.Effort has been put in developing methodologies for hair shape modeling, hair dynamics and hair rendering.A user friendly interface controlled by a haptic device facilitates designer's interactivity with the hairstyles.Furthermore, designer's visualization is enhanced by using real time animation and interactive rendering.Animation is done using a modified Free Form Deformation (FFD) technique that has been effectively adapted to various hairstyles.Hair Rendering is performed using an efficient scattering based technique, displaying hair with its various optical effects.

Typical real world machine control tasks have some characteristics which makes them difficult to solve: Their state spaces are high-dimensional and continuous, and it may be impossible to reach a satisfying target state by exploration or human control. To overcome these problems, in this paper, we propose (1) to use radial basis functions for value function approximation in continuous space reinforcement learning and (2) the use of learned inverse projection functions for state space exploration. We apply our approach to path planning in dynamic environments and to an aircraft autolanding simulation, and evaluate its performance.

We present the results obtained from spectroscopic investigations and quantum chemical calculations of the interaction of anisole (methoxybenzene) with small water clusters. The experiments have been carried out using resonant two-photon ionization (R2PI) and IR-UV double-resonance vibrational spectroscopy (IR/R2PI) in the region of the OH stretches. Apart from the vibrational spectra of the water moiety in the clusters, their intermolecular vibrations in the electronically excited S1 state are identified by IR/R2PI hole burning spectroscopy and assigned according to the vibrations calculated for the S1 state and compared with the vibrations calculated for the S0 state. The calculations for the S0 state were carried out at the second order Møller-Plesset level of theory using both the 6-31+G* and aug-cc-pVDZ basis sets and for the S1 state at the configuration interaction singles (CIS) level with the 6-31+G* basis set. In the electronic ground state (S0), the interaction of a water monomer to anisole is mediated through its oxygen atom, and that of a water dimer both through the oxygen atom (σ type of interaction) and the arene ring (π type of interaction). Thus in contrast to the interaction of fluorinated benzenes with water clusters, wherein a conformational transition from an in-plane σ to a on-top π bonding emerges starting with a water trimer, this conformational transition appears in case of anisole already with a water dimer. In the excited state (S1) of the investigated systems, there is a pronounced weakening of the interaction of the water cluster with the aromatic chromophore, which is also responsible for the blue shift of the electronic transitions. Consequently, the structures of the complexes of anisole with a water monomer or dimer are very different in both states. The weakening results from a diminished electron density of the oxygen atom and of the π system of anisole in the excited state. The calculated binding energies of the ground-state conformers indicate that these small water clusters are bound more strongly to anisole than to other π systems like benzene, toluene, fluorobenzene, and p-difluorobenzene. The many-body decomposition of the binding energy (S0) reveals that a progressive increase in the size of the water clusters results in a weakening of the σ O⋯H interaction and a concurrent strengthening of the π-H interaction. The complex containing a cyclic water trimer exhibits in the excited state also a π-type H-bonding interaction, but its stability emerges from a delocalization of the electron density from the water trimer to the anisole oxygen. Excepting the water dimer complexes, there is a good agreement between the calculated OH vibrational frequencies and the experimental IR spectra recorded for the ground state. The surprisingly good agreement of the calculated and the experimentally observed intermolecular modes in the excited state of these complexes provides convincing evidence that the experimental spectra emerge from structures similar to those predicted at the CIS/6-31+G* level.

The four-component matrix Dirac-Kohn-Sham (mDKS) implementation of EPR g- and hyperfine A-tensor calculations within a restricted kinetic balance framework in the ReSpect code has been extended to hybrid functionals. The methodology is validated for an extended set of small 4d(1) and 5d(1) [MEXn](q) systems, and for a series of larger Ir(II) and Pt(III) d(7) complexes (S = 1/2) with particularly large g-tensor anisotropies. Different density functionals (PBE, BP86, B3LYP-xHF, PBE0-xHF) with variable exact-exchange admixture x (ranging from 0% to 50%) have been evaluated, and the influence of structure and basis set has been examined. Notably, hybrid functionals with an exact-exchange admixture of about 40% provide the best agreement with experiment and clearly outperform the generalized-gradient approximation (GGA) functionals, in particular for the hyperfine couplings. Comparison with computations at the one-component second-order perturbational level within the Douglas-Kroll-Hess framework (1c-DKH), and a scaling of the speed of light at the four-component mDKS level, provide insight into the importance of higher-order relativistic effects for both properties. In the more extreme cases of some iridium(II) and platinum(III) complexes, the widely used leading-order perturbational treatment of SO effects in EPR calculations fails to reproduce not only the magnitude but also the sign of certain g-shift components (with the contribution of higher-order SO effects amounting to several hundreds of ppt in 5d complexes). The four-component hybrid mDKS calculations perform very well, giving overall good agreement with the experimental data.