Characterization of the Existence of an Enriched Linear Finite Element Approximation Using Biorthogonal Systems

The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enriched element we derive simple explicit formulas for the basis functions, and show that the approximation error can be written as the sum of the error of the (non-enriched) element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L² norm. We also give an alternative approach to estimate the approximation error, which relies on an appropriate use of the Poincaré inequality.     

A Class of Functional Equations on a Locally Compact Group

Let G be a locally compact group not necessarily unimodular.Let µ be a regular and bounded measure on G. We study,in this paper, the following integral equation, E(µ) This equation generalizes the functional equation for sphericalfunctions on a Gel'fand pair. We seek solutions in the spaceof continuous and bounded functions on G. If is a continuousunitary representation of G such that (µ) is of rank one,then tr((µ)(x)) is a solution of E(µ). (Here, trmeans trace). We give some conditions under which all solutionsare of that form. We show that E(µ) has (bounded and)integrable solutions if and only if G admits integrable, irreducibleand continuous unitary representations. We solve completelythe problem when G is compact. This paper contains also a listof results dealing with general aspects of E(µ) and propertiesof its solutions. We treat examples and give some applications.     

Measurement and prediction of solid–liquid phase equilibria for diamine + n-heptane, or cyclohexane

Solid–liquid equilibria (SLE) of N,N,N′,N′-tetramethylethylenediamine, 1,4-dimethylpiperazine and N,N-dimethylaniline+n-heptane or cyclohexane mixtures were measured by a static method. It was found that all systems are simple eutectic systems. Group contribution models have proved fairly successful in predicting SLE, however, the presence of intramolecular effects (ring effect, proximity effect) renders the widely used empirical methods quite inaccurate. However, in this work, the experimental phase diagrams compared satisfactorily with group contribution models (DISQUAC) and also modified UNIFAC (Dortmund version) predictions.     

Theoretical investigation of the low-lying electronic states of the alkaline hydride BeH2+ ion: Potential energy curves, spectroscopic constants, vibrational levels, and transition dipole functions

The structure and spectroscopic properties of the alkaline hydride BeH²⁺ ion have been investigated using an ab initio approach based on nonempirical pseudopotentials and parameterized l-dependent polarization potentials. The adiabatic potential energy curves and their spectroscopic constants for the ground and seventeen excited electronic states, dissociating into Be⁺(2s, 2p, 3s, 3p, 3d, 4s, 4p, and 4d) + H⁺ and Be²⁺ + H(1s and n = 2), of ²??⁺, ²??, and ²?? symmetries have been determined. As no experimental data are available, our results are discussed and compared with the few existing theoretical calculations. A very good agreement has been found with the previous theoretical data for the ground state; however many potential energy curves for the higher excited states are presented here, for the first time. Numerous avoided crossings between electronic states for ²??⁺ and ²?? symmetries have been localized and analyzed. Their existence is related to the interaction between the electronic states and to the charge transfer process between the two ionic systems Be²⁺H and Be⁺H⁺. In addition, we have calculated the vibrational energy level spacings of the bound electronic states. Furthermore, the adiabatic transition dipole functions from the X ²??⁺ and 2²??⁺ states to the higher excited states of ²??⁺ and ²?? symmetries have been evaluated and compared with the available theoretical work. This study represents the necessary initial step towards the investigation of the charge transfer processes in collision between Be⁺-H⁺ and Be²⁺-H.     

Theoretical investigation of the molecular structure and transition dipole moments of the NaK low lying electronic states

The electronic structure and the spectroscopic constants of the low lying electronic states of the NaK⁺ ionic molecule have been determined through using an ab initio approach involving a non-empirical pseudopotential for the Na and K cores and core valence correlation correction. The potential energy of nearly 26 electronic states of ²Σ⁺, ²Π, and ²Δ symmetries has been calculated up to their dissociation limit Na(4d) + K⁺ and Na⁺ + K(6s). Their spectroscopic constants (R_e, D_e, T_e, ω_e, ω_eχ_e, and B_e) are derived and compared with the few available theoretical studies. A good agreement has been found for the ground state and few excited states with previous works. New potential energy curves were presented, for the first time, for the higher excited states. Numerous avoided crossing between electronic states of ²Σ⁺, ²Π symmetries have been localized and analyzed. Their existences are related to the charge transfer between the two ionic molecules Na⁺K and NaK⁺. Furthermore, we have determined the transition dipole moments for several states and analyzed the avoided crossings related to charge transfer between alkaline atoms.     

Convexity results and sharp error estimates in approximate multivariate integration

An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P.C. Hammer and of Simpson's rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

     

A Definiteness Theory for Cubature Formulae of Order Two

Let Ω ⊂ ℝ^d be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short) if it approximates the integral ∫_Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω. This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels. After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae.     

Neutron-proton pairing effect on some even-even rare-earth proton-rich nuclei

The neutron-proton pairing effect on some even-even rare-earth proton-rich nuclei is studied. It is taken into account, in the isovector case, within the framework of the generalized Bogoliubov-Valatin transformation, using Woods-Saxon single-particle energies.     

Viscosity modeling of the ternary system 1-methylnaphthalene +n-tridecane+2,2,4,4,6,8,8-heptamethylnonane up to 100 MPa

Abstract

Viscosity measurements of well-defined mixtures can be used to evaluate existing viscosity models. Recently, an extensive experimental study of the viscosity up to 100 MPa have been carried out for the ternary system composed of 1-methylnaphthalene+n-tridecane+2,2,4,4,6,8,8-heptamethylnonane. This system is a very simple, representation of some petroleum distillation cuts at 510 K. Three recently proposed viscosity models with a physical and theoretical background have been evaluated on this ternary system. The evaluated models are based on the rough hard-sphere scheme, the free-volume, and the friction theory. For all models, the predicted viscosities for the three pure compounds, the three binaries and the ternary are within or close to the experimental uncertainty (±2%)). satisfactory for most industrial applications. This work shows the potential extension of these viscosity approaches to real petroleum fluids. For comparison purposes results obtained with the well-known mixing laws (Grunberg-Nissan and Katti-Chaudhri) are also presented. Further, it has been found that the widely used LBC model can not predict the viscosity of this ternary system satisfactorily.     

Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents

Mathematical Notes - This paper deals with the existence of a solution of a problem involving Leray–Lions type operators in Sobolev spaces with variable exponent. The proofs of our main...