The nonparametric integral of the Calculus of Variations as a Weierstrass integral: Existence and representation

The definition and properties of an abstract and very general nonparametric integral of the Calculus of Variations is presented. In harmony with the Lewy-McShane approach, the nonparametric integral ∝ f, for set functions ? taking their values in a Banach space E, is defined in terms of its associated parametric integral. For the latter use is made of the abstract parametric integral proposed by Cesari in Rⁿ and then extended to Banach spaces by Breckenridge, Warner, and the authors. A condition (c) is shown to be relevant for the existence of the integral, and is preserved by the nonlinear operation f. Also, for f nonnegative, a Tonelli-type theorem is proved in the sense that the so defined Weierstrass integral ∝ f is always larger than or equal to the corresponding Lebesgue integral, and equality holds if and only if absolute continuity conditions hold. In the proof a suitable martingale is associated and a convergence theorem for martingales is applied. Applications to the calculus of variations will follow.     

A fuzzy approach to the transportation problem

The transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed. For the solution of the problem the technique of parametric programming is used. This makes it possible to obtain not only the maximizing solution (according to the Bellman-Zadeh criterion) but also other alternatives close to the optimal solution.     

Generalized roughening transition and its effect on the string tension

The well-understood roughening transition of an interface in the d = 3 Ising model implies an essential singularity in the string tension of the dual Z₂ gauge model. The roughening transition corresponds to the delocalization of the string due to strong long-wavelength fluctuations, and this reformulation can be generalized to other gauge groups and to d = 4 also. It is not a deconfining transition - it is expected to occur deep in the confining region - but its presence would raise serious questions about the continuation of strong coupling expansions of the tension beyond this point. In this paper predictions on the roughening transition are confronted with the available information on the string tension for different gauge groups in three dimensions.     

Magnetic structure of the canted 1D-antiferromagnet NaBaFe2F9

NaBaFe₂F₉ crystallizes with the Ba₂CoFeF₉ structural type (space group P2₁/n, Z = 4). In this structure, infinite cis double [M₂F₉]^3? chains of corner sharing octahedra are isolated one from each other by sodium and barium ions. At 4.5 K, the cell parameters (Å) are found to be: a = 7.3236(3), b = 17.4525(7), c = 5.4586(2), β = 91.840(3)°. Antiferromagnetic interactions dominate but, below T_N ? 19 K, a parasitic ferromagnetic component appears with σ_r(4.5K) = 0.03(1) μ_B·mole^?.The magnetic structure (C_xF_yC_z mode) was foreseen by the macroscopic theory of Bertaut and established from neutron powder diffraction. Using the Rietveld profile refinement method, the 4.5 K spectrum was analysed at $\text{λ =}\text{1.909}\text{A}\text{?}\text{(R}{}_{\mathrm{<mtext>Profile</mtext>}}\text{=}\text{0.093}\text{, R}{}_{\mathrm{<mtext>Nucl</mtext>}}\text{=}\text{0.053}\text{, R}{}_{\mathrm{<mtext>Mag</mtext>}}\text{=}\text{0.126}\text{)}$ and the 4.5–35K difference spectrum at $\text{λ =}\text{2.518}\text{A}\text{?}\text{(R}{}_{\mathrm{<mtext>Mag</mtext>}}\text{=}\text{0.049}\text{, R}{}_{\mathrm{<mtext>Profile</mtext>}}\text{=}\text{0.092}\text{)}$. The magnetic moments on two Fe³⁺ sites, lie mainly in the (0 1 0) plane, approximately at 45° from the a- and c-axes. They form a pseudo G-type antiferromagnetic arrangement (μ(Fe1) = 3.60(4) μ_B;μ(Fe2) = 3.61(6)μ_B).     

Atmospheric electricity measurements in the upper air over Poona

Atmospheric electricity measurements made during the last decade at Poona using balloon-borne sondes are briefly reviewed. The vertical distribution of potential gradient in the free atmosphere follows theoretical values only during the monsoon months, when the air is comparatively free from dust. During the summer months, the potential gradient is constant above the exchange layer and this is associated with the presence of the deep, dense dust layer that lies over the region and extends to over 10 km. into the atmosphere. The increase with height of the potential gradient above the 10 km. level during winter is shown to be closely related to the presence of the subtropical jet stream, which presumably transports electric charges to the upper levels over Poona.     

Dependence of calculated collective nuclear properties on the form of the potential energy and inertia tensor

A systematic study of the dependence of the collective quantities (energies and matrix elements of the E2, M1 and E0 moments) on the form of approximations to the potential energy V and the inertia tensor B is performed. Various approximations used up to now are tested. Macroscopic-microscopic values for V and cranking results for B are taken as a reference. The collective quantities are calculated by solving the Schrödinger equation corresponding to the collective Bohr hamiltonian. The contribution of all nucleons is explicitly taken into account; no renormalization factors are used. Spherical, transitional and deformed even-even nuclei are considered. The quality of various approximations for V and B used in the boson-expansion method is discussed. Large effects of the microstructure of the inertia tensor B are obtained and commented on.     

Thermal stability of polyurethanes

Summary The combined application of thermogravimetry reactiongas chromatography and gel-permeation chromatography permits to follow the heat degradation of polyurethane polymers in inert gas, air and water-saturated environment. The examinations give information on the rate of thermal degradation, the individual volatile degradation components, the critical points of the polymer chains and on the change of their molecular-weight distribution. Gas chromatographic examinations also permit the identification of the chain-extending components of different types of polyurethanes.     

Modification and minor mechanization of instruments in the spectroscopic laboratory

Journal of Applied Spectroscopy -     

3-O-Methyl-6-desoxy-D-allose. Desoxyzucker, 38. Mitteilung

3-O-Methyl-6-deoxy-D -allose (V) and its methyl β-D -pyranoside (IV) have been synthesized and obtained in a crystalline state, although in low yield, starting from methyl 6-deoxy-β-D -allopyranoside (I) via the 2,4-di-O-benzoyl derivative II.