Correlation of the position of the bond functions in H-X with electronegativity

S type Gaussian bond functions are optimized for HF, H₂O, NH₃, and CH₄. The optimization is carried out with respect to the exponent and position in the H-X bond. The position is found to correlate well with the electronegativity of Pauling and Allred-Rochow.     

A six-dimensional quadrature procedure

An account is given of the use of Gaussian quadrature product formulae in the evaluation of certain six-dimensional, two-centre integrals involving one-electron Green's functions. These integrals occur in a new molecular variational principle recently proposed by Hall, Hyslop and Rees [1] from which an approximate energy may be derived which can be shown to be at least as good as that obtained from the Rayleigh-Ritz principle. Reductions in computing time are realized by removing certain singularities using a subtraction technique and also by using an empirically determined Richardson-type extrapolation formula.This paper was presented during the session on numerical integration methods for molecules of the 1970 Quantum Theory Conference in Nottingham. It has been revised in the light of the interesting discussion which followed.     

High temperature enthalpy and heat capacity of GaN

The heat capacity and the heat content of gallium nitride were measured by calvet calorimetry (320-570 K) and by drop calorimetry (670-1270 K), respectively. The temperature dependence of the heat capacity in the form C_pm=49.552+5.440×10⁻³T−2.190×10⁶T⁻²+2.460×10⁸T⁻³ was derived by the least squares method. Furthermore, thermodynamic functions calculated on the basis of our experimental results and literature data on the molar entropy and the heat of formation of GaN are given.     

Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF: and byDom(F) its domain, it is the set {uεL ^p(T,E):ū(t)=f(u),t)εL ¹(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL ^p(T, E).     

Ground-state energy of a quantum chain with competing interactions

We show rigorously that the ground state of a quantum chain with competing ferromagnetic nearest and antiferromagnetic next nearest interactions undergoes a transition from ferromagnetic to helical type, in the isotropic case, for a certain value of the relevant ratio of coupling constants. Boundaries of the phase diagram are also determined in the anisotropic case. The stability of a special quantum state (corresponding to a classical modulated phase of =/3) is analyzed by an extension of Holstein-Primakoff arguments, along a line of constant ratio of couplings, showing in particular a sequence of (instability) gaps. Finally, a natural adaptation of a variational wave function due to Huse and Elser is used to study several portions of the phase diagram, with very good agreement with previous theoretical results.     

On the complete solution of ∈y″=y 3

In Ref. 1, the author claimed that the problem y=y ³ is soluble only for a certain range of the parameter . An analytic approach, as adopted in the following contribution, reveals that a unique solution exists for any positive value of . The solution is given in closed form by means of Jacobian elliptic functions, which can be numerically computed very efficiently. In the limit 0+, the solutions exhibit boundary-layer behavior at both endpoints. An easily interpretable approximate solution for small is obtained using a three-variable approach.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Characterization of measures by potentials

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f _E f(‖x?a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l _∞ ⁿ and wheref(t)=|t| ^p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ _∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ _∞) is positive definite only iff is a constant function.     

Rational choice based on vague preferences

This paper studies vague preferences. In contrast to the concept of fuzzy preferences, the more general notion of vague preferences does not entail the assumption that the different degrees of preference are completely ordered. It is shown that the more general concept of a vague preference order arises very naturally in the context of aggregating a set of (exact) weak orders. Furthermore, necessary and sufficient conditions for the rationalizability of an exact choice function by means of a vague preference order are given. It turns out that in the general case these conditions are weaker than in the case where the degrees of preference are completely ordered. Consequently, there exist choice functions which are rationalizable by a general vague preference order but which are not rationalizable by a fuzzy preference order, or more generally by any vague preference order which induces a complete ranking of the degrees of preference. Examples of such choice functions are provided.This is a revised version of a discussion paper which was written at the University of Karlsruhe. I am grateful to Georg Bol and an anonymous referee for valuable comments. The research has been made possible by a fellowship of the DFG (Deutsche Forschungsgemeinschaft) which is also gratefully acknowledged.