On the final steps of Newton and higher order methods

The Newton method is one of the most used methods for solving nonlinear system of equations when the Jacobian matrix is nonsingular. The method converges to a solution with Q-order two for initial points sufficiently close to the solution. The method of Halley and the method of Chebyshev are among methods that have local and cubic rate of convergence. Combining these methods with a backtracking and curvilinear strategy for unconstrained optimization problems these methods have been shown to be globally convergent. The backtracking forces a strict decrease of the function of the unconstrained optimization problem. It is shown that no damping of the step in the backtracking routine is needed close to a strict local minimizer and the global method behaves as a local method. The local behavior for the unconstrained optimization problem is investigated by considering problems with two unknowns and it is shown that there are no significant differences in the region where the global method turn into a local method for second and third order methods. Further, the final steps to reach a predefined tolerance are investigated. It is shown that the region where the higher order methods terminate in one or two iteration is significantly larger than the corresponding region for Newton’s method.     

Diameter Two Properties,Convexity and Smoothness

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties.We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space \({X}\): “For every slice \({S}\) of \({B_X}\) and every norm-one element \({x}\) in \({S}\), there is a point \({y \in S}\) in distance as close to 2 as we want.” Spaces with this property are shown to have non-smooth bidual.     

An investigation of the quantum chemical description of the ethylenic double bond in reactions: II. Insertion of ethylene into a titanium–carbon bond

Insertion of ethylene into the Ti–methyl bond in TiH₂CH⁺₃ is chosen as a model reaction for investigating the performance of a range of contemporary quantum chemical models in polymerization studies. Basis set effects are investigated at the self-consistent-field level, covering Hartree–Fock, pure DFT, and hybrid DFT. In agreement with findings in part I of this study, the basis set sensitivity of ethylene is shown to introduce a bias in computed energetics, amounting to 2–3 kcal/mol when DZP bases are used to compute the overall heat of monomer insertion. The geometry of stationary points relevant to the insertion reaction is determined using hybrid density functional theory. Based on these structures, the energy profile of the insertion reaction is computed using a range of popular quantum chemical approximations. The methods include Hartree–Fock and Møller–Plesset (MP) perturbation theory up through the fourth order in spin-restricted, spin-unrestricted, and spin-projected formalisms. Furthermore, configuration-interaction-based methods are included, of which the top level method is singly and doubly excited coupled clusters with a perturbative estimate of the contribution from triply excited configurations added [CCSD(T)]. The performance of the methods just mentioned, as well as three pure density functional and three hybrid density functional methods, are compared with respect to “best” relative energies, defined through extrapolation of CCSD(T) correlation energies according to the PCI scheme of Siegbahn and coworkers. Even though the MP series show poor convergence, spin-projected MP2, as well as two pure DFT methods (BPW91, BP86) and PCI-78 based on the MCPF method, show similar and very good agreement with best relative energies for the insertion reaction. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 947–960, 1998     

Parity nonconservation contribution to the nuclear magnetic resonance shielding constants of chiral molecules: a four-component relativistic study

A systematic four-component relativistic study of the parity nonconservation (PNC) contribution to the (isotropic) NMR shielding constants of chiral molecules is presented for the P enantiomers of the series H(2)X(2) (X=(17)O,(33)S,(77)Se,(125)Te,(209)Po). The PNC contributions are obtained within a linear response approach at the Hartree-Fock level. A careful design of the basis sets is necessary. The four-component relativistic results based on the Dirac-Coulomb Hamiltonian are compared with the nonrelativistic Levy-Leblond results and those obtained by the spin-free modified Dirac Hamiltonian. The calculations confirm the nonrelativistic scaling law Z(2.4) of the PNC contribution with respect to nuclear charge Z. However, the calculations also show that the overall scaling is significantly modified by relativistic effects. The scalar relativistic effect scales as Z(4.7) for the selected set of molecules, whereas the spin-orbit effect, of opposite sign, scales better than Z(6) and completely dominates the PNC contribution for the heaviest elements. This opens up the intriguing possibility of the experimental observation of PNC effects on NMR parameters of molecules containing heavy atoms. The presented formalism is expected to be valuable in assisting the search for suitable candidate molecules.     

Effects of photon field on heat transport through a quantum wire attached to leads

We theoretically investigate photo-thermoelectric transport through a quantum wire in a photon cavity coupled to electron reservoirs with different temperatures. Our approach, based on a quantum master equation, allows us to investigate the influence of a quantized photon field on the heat current and thermoelectric transport in the system. We find that the heat current through the quantum wire is influenced by the photon field resulting in a negative heat current in certain cases. The characteristics of the transport are studied by tuning the ratio, $?{\omega }_{\gamma }/k_{\mathrm{B}}\mathrm{\Delta }T$, between the photon energy, $?{\omega }_{\gamma }$, and the thermal energy, $k_{\mathrm{B}}\mathrm{\Delta }T$. The thermoelectric transport is enhanced by the cavity photons when $k_{\mathrm{B}}\mathrm{\Delta }T>?{\omega }_{\gamma }$. By contrast, if $k_{\mathrm{B}}\mathrm{\Delta }T<?{\omega }_{\gamma }$, the photon field is dominant and a suppression in the thermoelectric transport can be found in the case when the cavity-photon field is close to a resonance with the two lowest one-electron states in the system. Our approach points to a new technique to amplify thermoelectric current in nano-devices.     

Bayesian Assessment of Availabilities and Unavailabilities of Multistate Monotone Systems

In the present paper we consider a multistate monotone system of multistate components. Following a Bayesian approach, the ambition is to arrive at the posterior distributions of the system availabilities and unavailabilities to the various levels in a fixed time interval based on both prior information and data on both the components and the system. We argue that a realistic approach is to start out by describing our uncertainty on the component availabilities and unavailabilities to the various levels in a fixed time interval, based on both prior information and data on the components, by the moments up till order m of their marginal distributions. From these moments analytic bounds on the corresponding moments of the system availabilities and unavailabilities to the various levels in a fixed time interval are arrived at. Applying these bounds and prior system information we may then fit prior distributions of the system availabilities and unavailabilities to the various levels in a fixed time interval. These can in turn be updated by relevant data on the system. This generalizes results given in Natvig and Eide (Scand J Statist 14:319?C327, 1987) considering a binary monotone system of binary components at a fixed point of time. Furthermore, considering a simple network system, we show that the analytic bounds can be slightly improved by straightforward simulation techniques.     

Sparsity of the average curvature information matrix

We present a theorem regarding the average curvature properties of partially separable functions which need not be differentiable or continuous. This has implications for derivative-free optimization methods which make use of average curvature information to select the set of search directions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear interference in a mean-field quantum model

Using similar nonlinear stationary mean-field models for both a 2D axisymmetricalBose-Einstein Condensate and an electron pair in a parabolic trap, we propose to describethe original many-particle ground state as a one-particle mixed state (in contrast to apure state), i.e. as a statistical ensemble of several one-particle quantum states. Thesequantum states are the eigenfunctions of the corresponding stationary nonlinearSchrödinger equation (hence called “nonlinear eigenstates”). Due to their nonlinearity,they are not orthogonal. Therefore, taking the simple example of a two-level system, weshow that each of these two nonlinear eigenstates |i? and|j? occurs with a probability (or statistical weight) that isdefined by their non-orthogonality ?i|j? 0. We givethe corresponding density matrix. We search for physical grounds in the interpretation ofour two main results, namely, a quantum-classical nonlinear transition and theinterference between two “nonlinear eigenstates”.     

Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems

In the condensed liquid phase, both single- and multicomponent Lennard–Jones (LJ) systems obey the “hidden-scale-invariance” symmetry to a good approximation. Defining an isomorph as a line of constant excess entropy in the thermodynamic phase diagram, the consequent approximate isomorph invariance of structure and dynamics in appropriate units is well documented. However, although all measures of the structure are predicted to be isomorph invariant, with few exceptions only the radial distribution function (RDF) has been investigated. This paper studies the variation along isomorphs of the nearest-neighbor geometry quantified by the occurrence of Voronoi structures, Frank–Kasper bonds, icosahedral local order, and bond-orientational order. Data are presented for the standard LJ system and for three binary LJ mixtures (Kob–Andersen, Wahnström, NiY2). We find that, while the nearest-neighbor geometry generally varies significantly throughout the phase diagram, good invariance is observed along the isomorphs. We conclude that higher-order structural correlations are no less isomorph invariant than is the RDF.