Ni-Mn-Ga magnetic shape memory alloys (MSMAs) tend to undergo a large deformation upon the application of a magnetic field. This deformation is attributed to twin boundary motion in the martensitic phase. In an effort to harness the shape memory effect for use in sensors, actuators, and micro-devices, the behavior of Ni-Mn-Ga thin films is attracting attention. Substrate curvature measurements were done with Ni-Mn-Ga films with a thickness of 2.0 μm sputter-deposited on Si(100) wafer having amorphous 500 nm thick SiNx buffer layer. During the wafer bow curvature measurements, stress levels of 0.65 GPa were attained. The martensitic transformation is manifested by a stress-temperature hysteretic loop. Measurements of magnetization curves were carried out on Ni-Mn-Ga films with thickness between 0.5 and 3.0 μm. A change of the magnetization behavior from the easy-plane type for thin films to the out-of-plane easy-axis type for thick films is observed. This effect is caused by the interplay between different contributions to the overall anisotropy of film.

Constant-stress compressive creep tests were carried out on an Al-rich a2 single-phase material, which had equiaxed-grains of 60μim in grain size, at 1050∼1250 K under 100∼500MPa. The type of the primary creep stage and the microstructures developed during creep depend greatly on the creep condition. The minimum creep-rate, however, can be represented by one set of parameters over the whole range of experimental condition. The stress exponent is 5.0±0.2 and the (modulus-compensated) activation energy is 360 ± 10kJ/mol. The Dorn-type plot of the minimum creep rate reveals that the normalized creep strength of fine-grained Ti-34mol%Al is not greatly different from that of disordered solid-solution hardened alloys.

The electric capacity of a conductor in the 3-dimensional euclidean space is defined as the ratio of a positive charge given to the conductor and the potential on its surface. The notion of capacity was defined mathematically first by N. Wiener [7] and developed by C. de la Vallée Poussin, O. Frostman and others. For the history we refer to Frostman’s thesis [2]. Recently studies were made on different definitions of capacity and related notions. We refer to M. Ohtsuka [4] and G. Choquet [1], for instance. In the present paper we shall investigate further some relations among various kinds of capacity and related notions. A part of the results was announced in a lecture of the author in 1962.

Si X est un ensemble mesurable au sens de Lebesgue dans un espace euclidien En et Y est un ensemble mesurable au sens de Lebesgue dans un autre espace euclidien Em , alors X × Y est mesurable dans En+m= Enx Em et sa mesure est égale au produit de la mesure de X et celle de Y. Mais il n’existe pas de telle relation simple ni pour la capacité d’ordre général ni pour la mesure de Hausdorff de dimension générale. Dans le présent mémoire nous essayons d’évaluer la capacité des ensembles produits au moyen de la capacité ou de la mesure de Hausdorff des ensembles composants.

La question posée par Noshiro dans [5] pour les ensembles d’accumulation relatifs à des fonctions pseudo-analytiques classiques a été résolue par (T.) Yosida [12] premièrement et ensuite, le présent auteur [8] a étendu son résultat (théorème 1 de [12]) au cas où un nombre fini ou dénombrable de surfaces de Riemann, qui sont des surfaces de recouvrement, sont transformées dans une autre surface de Riemann par une application continue à dérivées partielles continues sauf en un ensemble fermé de mesure linéaire nulle, ayant pour image un ensemble de mesure linéaire également nulle; le quotient de dilatation n’y est pas nécessairement borné mais subit certaine condition, et les ensembles d’accumulation y sont définis pour les éléments frontières de Kerékjartô-Stoïlow. Une généralisation d’un théorème étoilé de Gross obtenue dans [7] a été utilisée dans sa démonstration. L’auteur a continué son étude sur ce théorème étoilé et un de ses résultats établis dans [9] lui permettra d’améliorer les résultats de [8] dans le présent mémoire. L’idée fondamentale est la même que dans [8] mais nous la récrirons, en nous reportant à [8] le moins possible.

D’abord nous définissons les ensembles de Cantor généralisés. Soient k1, k2 , … des nombres entiers supérieurs à 1 et soient p1, p2 , … des nombres finis quelconques également supérieurs à 1. On pose lq =l/(kqpq ). Soit I un intervalle de longueur d > 0. On enlève de I (kq — l) intervalles de même longueur tels qu’il reste kq intervalles de même longueur lqd. On appelle cette opération la q-opération appliquée à I. On commence par appliquer l’1-opération à [0,1], on applique la 2-opération à chacun des intervalles qui restent, puis on applique la 3-opération à chacun des intervalles qui restent, et ainsi de suite. On appellera l’ensemble limite restant un ensemble de Cantor généralisé dans E 1 et le notera F = F(kq, pq). Notre définition dans En = {(x 1, …, xn )}, l’espace euclidien à n dimensions (n ≧ 2), est la suivante: Soit un ensemble de Cantor généralisé défini sur l’axe de xj ; nous appellerons l’ensemble produit F = F1 × … × Fn un ensemble de Cantor généralisé dans En (n ≧ 2). II sera appelé symétrique si F1 = … =Fn .

Let f(z) be a nonconstant analytic transformation of the unit circle U : ∣z∣ < 1 into a Riemann surface ℜ. As an extension of a classical theorem of F. and M. Riesz, the author proved in Theorem 3.4 of [2] that if the image of U is relatively compact in ℜ and has universal covering surface of hyperbolic type, and if, at every point of a set on ∣z∣ = 1 of positive inner linear measure, there terminates a curve along which f(z) has limit, then the set of such limits has positive inner logarithmic capacity. This theorem was followed by the first proposition in Kuramochi [1], which asserts that, if ℜ has a null boundary and the image of U excludes a set of positive logarithmic capacity on ℜ and if, at every point of a set E on ∣z∣ = 1, there terminates a curve along which f(z) has limit in the union of a set of inner logarithmic capacity zero on ℜ and the boundary components of ℜ then the inner linear measure of E is zero.

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined by

Theorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

Montel [10] proved in 1912 the following theorem: Let w = f(z) be an analytic function in the horizontal strip B : 0 < x < + ∞, 0 < y < 1 (z = x + iy) which is continuous on 0 < x < + ∞, 0 ≦ y < 1 and omits at least two values. If f (x) converges to a value w0 as x → + ∞, then f(z) converges to W0 as z tends to in 0 < x < + ∞, 0 ≦ y < 1 − ε for any ε such that 0 < ε < 1.

Nous continuerons l’étude du premier théorème étoilé de Gross, faite dans le mémoire de l’auteur: Théorèmes étoilés de Gross et leurs applications, Ann. Inst. Fourier, 5 (1955), pp. 1-28; ce mémoire sera cité avec l’abréviation [A]

M. Brelot [1] has shown that if u(z) is subharmonic in an open set D in the z-plane with boundary C and is bounded from above in a neighborhood of a boundary point z 0, which is contained in a set E ⊂ C of inner harmonic measure zero with respect to D, and such that z 0 is a regular point for Dirichlet problem in D, then

(1) .

The boundary components of an abstract Riemann surface were defined by B. v. Kérékjértó [7] and utilized in the book [14] written by S. Stoïlow. It is the purpose of the present paper to investigate their images under conformal mapping and to solve the Dirichlet problem with boundary values distributed on them.

The following relation was set up in [5] for an open covering Riemann surface ℜ with positive boundary over an abstract Riemann surface

(1)

when the universal covering surface of the projection is not of hyperbolic type when is of hyperbolic type this relation is reduced to

(2)

1. Let be an abstract Riemann surface in the sense of Weyl-Radó, and an open covering surface over . If a curve C = {P(t);0≦t<1} on tends to the ideal boundary of but its projection terminates at an inner point of as t→1, we shall say that C determines an accessible boundary point (which will be abbreviated by A.B.P.) of relatively to . The set of all the A.B.P.S of relative to will be called accessible boundary (relative to ) and denoted by 3i() or by (, ). Throughout in this paper () will be supposed to be non-empty.

Let D be an open set in the 2-plane, C its boundary, z 0 a point on C, and f(z) a one-valued meromorphic function in D.

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.