Conjugate Directions without Linear Searches

A modified form of the Quasi-Newton family of variable metricalgorithms used in function minimization is proposed that hasquadratic termination without requiring linear searches. Mostmembers of the Quasi-Newton family rely for quadratic terminationon the fact that with accurate linear searches the directionsgenerated, form a conjugate set when the function is quadratic.With some members of the family the convergence of the sequenceof approximate inverse Hessian matrices to the true inverseHessian is also stable. With the proposed modification the samesequence of matrices and the same set of conjugate directionsare generated without accurate linear searches. On a quadratic function the proposal is also related to a suggestionby Hestenes which generates the same set of conjugate directionswithout accurate linear searches. Both methods therefore findthe minimum of an n dimensional quadratic function in at mostn+2 function and gradient calls. On non-quadratic functions the proposal retains the main advantagesclaimed for both the stable Quasi-Newton and Hestenes approaches.It shows promise in that it is competitive with the most efficientunconstrained optimization algorithms currently available.     

Conjugate Gradient Algorithms: Quadratic Termination without Linear Searches

This paper studies the behaviour of a family of conjugate gradientoptimization algorithms, of which the best known is probablythat introduced in 1964 by Fletcher & Reeves. This familyhas the property that, on a quadratic function, the directionsgenerated by any member of the family are the same set of conjugatedirections providing that, at each iteration, an exact linearsearch is performed. In this paper a modification is introduced that enables thisset of conjugate directions to be generated without any accurateline searches. This enables the minimum of a quadratic functionto be found in, at most, (n+2) gradient evaluations. As themodification only requires the storage of two additional n-vectors,the storage advantage of conjugate gradient algorithms viz-?-vizvariable metric algorithms is maintained. Finally, a numerical study is reported in which the performanceof this new method is compared to that of various members ofthe unmodified family.     

Remarks on the signs of g factors in atomic and molecular Zeeman spectroscopy

This paper draws attention to the advantages that would be obtained by adopting a new convention for the sign of g factors that would make the g factor for electron spin a negative quantity (g ≈ ?2), rather than a positive quantity as generally adopted at present. The editors are aware that the proposal made in this paper concerning the conventional sign of the g factor for electron spin will be seen by some readers as controversial. We have nonetheless agreed to publish this paper in the hope that it will stimulate discussion. The editors would welcome comments on this proposal in the form of short papers, which they will then be happy to consider for publication together at a later date.

Various magnetic moments, associated with rotational, vibrational, nuclear spin, electron orbital and electron spin angular momenta, can contribute to the Zeeman effect in atoms and molecules. They are considered in this paper in the context of the effective Hamiltonian where relativistic and other corrections as well as the effects of mixing with other electronic states are absorbed in appropriate g factors. In spherically symmetric systems, the magnetic dipole moment arising from a specific angular momentum can be written as the product of three factors: the nuclear or Bohr magneton (which is positive), the g factor (which may be positive or negative), and the corresponding angular momentum (which is a vector). A convention is discussed, in which the sign of the g factor is positive when the dipole moment is parallel to its angular momentum and negative when it is antiparallel. This would have the advantage that it could be applied consistently in any situation. Such a choice would require the g factors for the electron orbital and electron spin angular momenta to be negative. This concept can easily be extended to the case of a general molecule where the relation between the dipole and angular momentum vectors has tensorial character.     

论邓-王定理

对王曾叙述了并证明了若干重要定理，本文给出在稍稍不同条件下的类似定理的更简单的证明，讨论这些定理的含义、作用并予以推广。     

A Unified Approach to Convergence Analysis of Discretization Methods for Volterra-typeEquations

This paper presents a general convergence analysis of numericalmethods for solving ordinary differential equations and non-linerVolterra integral and integrodifferential euqations. The conceptof analytic and discrete forms is introduced, Prolongation andrestriction operators reduce the problem of comparing the fundamentalforms. Integral inequalities and their discrete analogues proofof a collocation method for solving Volterra integral equationsof the second kind.     

The photodetachment spectrum of OHF−: the influence of vibration at a transition state

New ab initio potential energy functions for collinear OHF^? and OHF have been used with time-dependent wavepacket studies in a simulation of the photodetachment spectrum of the OHF^? anion. Franck—Condon excitation of OHF^? lies within the transition state region for the bimolecular reaction F + OH ? O + HF on the lowest triplet surface. The branching ratios for fragmentation of the excited OHF are such that the peaks in this spectrum may each be correlated with dissociation to a single vibrational state of the O + HF(v) or OH(v) + F products. The possible influence of vibrational excitation of the anion has been explored also. This is predicted to lead to marked changes in the spectra. Experimental observations of such spectra could be used in refinement of the potential as could an experiment in which the electron kinetic energy is detected in coincidence with the vibrational states of the molecular products or with the kinetic energy of the atomic products. The calculations revealed a number of metastable resonances on the OHF surface, although these are all at higher energies than are relevant to the photodetachment spectrum from the ground state anion.