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A finite net N of degree k, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line is simply a net of degree n+ 1, order n (Marshall Hall ). A loop of order n is essentially a net of degree 3, order n (Baer , Bates ). More generally, for , a set of k —2 mutually orthogonal n ⨯ n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose ) between affine planes and complete sets of orthogonal latin squares.
A heterophase polydomain structure has been recently discovered in BiFeO3 epitaxial ferroelectric films, which provides large electromechanical responses. In this work, the formation of such a microstructure is explained by theory of elastic domains. The thermodynamics of the heterophase polydomain microstructure is analyzed to predict the equilibrium volume fraction of domains at different film-substrate lattice misfits. Extrinsic mechanical and piezoelectric properties are discussed for the heterophase polydomains. It is shown that an applied electric field, which increases electrostatic interaction between domains, may lead to dramatic increase of piezo response. The results of this work are in good agreement with experimental data for BiFeO3.
Neutron diffraction was used to investigate the residual stress distribution in an axisymmetric A12O3-Ni joint bonded with a 40 vol%A12O3-60 vol%Ni composite layer. A series of measurements was taken along the axis of symmetry through the A12O3 and composite layers. It is shown that after taking into account the finite neutron diffraction sampling volume, both the trends and peak values of the experimentally determined strain distribution were in excellent agreement with calculations of a simple finite element model, where the rule-of-mixtures approach was used to describe the constitutive behavior of the composite interlayer. In particular, the predicted steep strain gradient near the interface was confirmed by the experimental data.
Single crystals of barium samarium titanium oxide Ba6−3xSm8+2xTi18O54 (x = 0.27) have been synthesized and studied using x-ray diffraction. Superstructure reflections, which cause a doubling of the cell along the short axis, were taken into account and the refinement was conducted in the orthorhombic space group Pnma. Unit cell parameters from single crystal x-ray diffraction were a = 22.289(1), b = 7.642(1), and c = 12.133(1) Å. Refinement on F resulted in R1 = 5.37% for 1410 Fo > 4σ with the thermal parameters of the Sm and Ba atoms refined anisotropically and the thermal parameters of the Ti and O atoms refined isotropically. The structure is made up of a network of corner sharing TiO6−2 octahedra creating rhombic (perovskite-like) and pentagonal channels. The two pentagonal channels are fully occupied by Ba atoms. The refinement suggests that one rhombic channel is fully occupied by Sm atoms (Sm3/Sm4), one rhombic channel is partially occupied by Sm atoms (100% Sm1/86.25% Sm5), and one rhombic channel is shared by BaySm atoms (59.25% Ba3/40.75% Sm2), resulting in a formula of Ba10.38Sm17.08Ti36O108 with Z = 1. The above site occupancies differ from the site occupancies previously reported in the literature for refinements conducted with the short axis approximately equal to 3.8 Å.
In 1993, a new beryllium bearing bulk metallic glass with the nominal composition Zr41.25Ti13.75Cu12.5Ni10Be22.5 was discovered at Caltech. This metallic glass can be cast as cylindrical rods as large as 16 mm in diameter, which permitted specimens to be fabricated with geometries suitable for dynamic testing. For the first time, the dynamic compressive yield behavior of a metallic glass was characterized at strain rates of 102 to 104/s by using the split Hopkinson pressure bar. A high-speed infrared thermal detector was also used to determine if adiabatic heating occurred during dynamic deformation of the metallic glass. From these tests it appears that the yield stress of the metallic glass is insensitive to strain rate and no adiabatic heating occurs before yielding.
In the course of preparing a book on group theory  with special reference to the Restricted Burnside Problem and allied problems I stumbled upon the concept of a dimension-linking operator. Later, when I lectured to the Third Summer Institute of the Australian Mathematical Society , G. E. Wall raised the question whether the dimension-linking operators could be made into a ring by introduction of a suitable definition of multiplication. The answer was easily found to be affirmative; the result wasthat the theory of dimen sion-linking operators became exceedingly simple.
A projective plane geometry π is a mathematical system composed of undefined elements called points and undefined sets of points (at least two in number) called lines, subject to the following three postulates:
(P1) Two distinct points are contained in a unique line.
(P2) Two distinct lines contain a unique common point.
(P3) Each line contains at least three points.
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