Work function of metals: Relation between theory and experiment

Theoretical approaches to calculation of work function within jellium model and the problem of extension of this model to include the lattice corrections to the work function are briefly discussed. Lattice corrections to the work function obtained from the experiment are estimated and compared with those calculated theoretically.

It is found that the mean value of the experimental lattice correction <δψ_hkl>_hkl compared to the mean work function is negligible. It is stated that the mean work function can be treated as a material constant characterizing a given metal, such as, e.g., binding energy.An expression for the dependence of jellium work function on r_s, valid in a metallic range of r_s, is given. A comparison between then theoretical and experimental results is presented and the role of correlation energy is examined. It is shown that more accurate approximations of the correlation energy than that given by Wigner's formula lead to a better agreement with experiment. A simple model is presented for explanation of work function changes on single crystal planes. Some recent results concerning the thermal dependence of work function are given. The dependence of the work function on the degree of coverage is discussed both for alkali and non-alkali atoms adsorption. Theoretical models are briefly reviewed and comparison between theory and experiment is made. A simple model is presented for explanation of the work function variation on rough planes in metallic non-alkali atoms chemisorption.     

Partial fractions expansion: a review of computational methodology and efficiency

Nine methods for expressing a proper rational function in terms of partial fractions are presented for the case where the denominator polynomial has been reduced to linear factors. Only those methods which are amenable to computation algorithms are considered. To the extent possible, Newton's divided difference formula is used to provide a uniform derivational tool. Each method is illustrated numerically. The efficiency of the methods are compared on the basis of the number of multiplications and divisions required in the computation.     

A linear filtering problem in complete correlated actions

In the last paper, the geometry of the Sz.-Nagy-Foia model for contraction operators on Hilbert spaces was used to advantage in several problems of multivariate analysis. The lifting of intertwining operators, one of the basic results from the Sz.-Nagy-Foia theory, is now recognized as the most adequate operatorial form of the deep classical results of the extrapolation theory. The labeling of the exact intertwining dilations given by [1]Acta Sci. Math. (Szeged) 40 9–32] and the recursive methods used there open a broad perspective for using the Sz.-Nagy-Foia model in multivariate filtering theory. In this paper, using the notion of correlated action (see [5 and 6] Rev. Roumaine Math. Pures Appl. 23, No. 9 1393–1423]) as a time domain, a linear filtering problem is formulated and its solution in terms of the coefficients of the analytic function which factorizes the spectral distribution of the known data and the coefficients of an analytic function which describes the cross correlations is given. In some special cases it is shown that the filter coefficients can be determined using recursive methods from the intertwining dilation theory, of the autocorrelation function of the known data and an intertwining operator, interpreted as the initial estimator given by the prior statistics.     

A class of loss functions of catenary form

The catenary form of loss function is considered in the framework of Bayesian decision theory. The mathematical tractability of this form seems to be unrecognized; it contains quadratic loss as a limiting case. For various probability distributions expressions are given for posterior analysis, and limiting properties are investigated.     

A combined penalty function and gradient projection method for nonlinear programming

A new programming algorithm for nonlinear constrained optimization problems is proposed. The method is based on the penalty function approach and thereby circumyents the necessity to maintain feasibility at each iteration, but it also behaves much like the gradient projection method. Although only first-order information is used, the algorithm converges asymptotically at a rate which is independent of the magnitude of the penalty term; hence, unlike the simple gradient method, the asymptotic rate of the proposed method is not affected by the ill-conditioning associated with the introduction of the penalty term. It is shown that the asymptotic rate of convergence of the proposed method is identical with that of the gradient projection method.Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GK-16125.     

About membership functions estimation

A logical approach to fuzzy sets method originated by Giles is developed. The infinitely many-valued logic t_ω is taken as basic. We accept, it is correct to use the strong conjunction by the logical analysis of the summation of fuzzy items. Under some broad conditions it is proved. that the sum of many fuzzy variables is a variable whose membership function is approximately equal to $\text{?(x) =}\text{max}\text{{1 ?}\text{1}\text{2}\text{c(x ? α)}{}^2\text{, 0}}$, where a and c are some constant parameters. A method of estimation of the unknown parameters is developed in a general case. The proposed fuzzy method coincides with the method of maximum likelihood if used in problems of classical mathematical statistics.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

Milling of Organic Solids in a Jet Mill. Part 2: Checking the Validity of the Predicted Rate of Breakage Function

The particle size distribution of fine chemicals in the solid state, like active pharmaceutical ingredients, is often a critical parameter. To achieve the desired particle size distribution, milling of such materials is usually the method of choice. Since these chemicals are often scarcely available, experimental optimization of milling is not possible. Therefore, a model to predict the milling conditions has been developed. The model estimates the rate of breakage function, and needs mechanical properties like hardness and yield strength as input to calculate the rate of breakage function. This paper attempts to check the validity of the model by a series of experiments. A comparison of the experimental results with the outcomes of the model using five different model compounds has been performed. It appears that the rate of breakage function can be estimated by: The model is able to rank the compounds by degree of fracture as an effect of milling. It was also possible to perform a quantitative prediction of the impact of milling pressure on the milling behavior. Finally, it appeared that the prediction of the large particles in the distribution was significantly better than small ones. Because the oversized material is usually the most critical parameter, the conclusion is that the model has acceptable practical applicability.     

Extended structures and new infinite-dimensional hidden symmetries for the Einstein–Maxwell-dilaton-axion theory with multiple vector fields

A so-called extended elliptical-complex (EEC) function method is proposed and used to further study the Einstein–Maxwell-dilaton-axion theory with p vector fields (EMDA-p theory, for brevity) for . An Ernst-like matrix EEC potential is introduced and the motion equations of the stationary axisymmetric EMDA-p theory are written as a so-called Hauser–Ernst-like self-dual relation for the EEC matrix potential. In particular, for the EMDA-2 theory, two Hauser–Ernst-type EEC linear systems are established and based on their solutions some new parametrized symmetry transformations are explicitly constructed. These hidden symmetries are verified to constitute an infinite-dimensional Lie algebra, which is the semidirect product of the Kac–Moody algebra and Virasoro algebra (without centre charges). These results show that the studied EMDA-p theories possess very rich symmetry structures and the EEC function method is necessary and effective.