About a minimax theorem of Matthies,Strang and Christiansen

We give a new minisup theorem for noncompact strategy sets. Our result is of the type of the Matthies-Strang-Christiansen minimax theorem where the hyperplane should be replaced by any closed convex set. As an application, we derive a slight generalization of the Matthies-Strang-Christiansen minimax theorem.     

Geometry of optimality conditions and constraint qualifications: The convex case

The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem. In addition, the badly behaved set of constraints, i.e. the set of constraints which causes problems in the Kuhn—Tucker theory, is isolated and a computational procedure for checking whether a feasible point is regular or not is presented.This research was supported by the National Research Council of Canada and le Gouvernement du Quebec and is part of the author's Ph.D. Dissertation done at McGill University, Montreal, Que., under the guidance of Professor S. Zlobec.     

Crystal structures and phase transition in the system SrTiO3-La2/3TiO3

The distribution of A-site cations in the perovskite system La_xSr_1−3x/2TiO₃ depends on the concentration of La³⁺ ions and associated vacancies. For small x (x?0.2), the substitutions are expected to be random. For x?0.55, the cations are ordered in such a way that successive layers of A-sites are occupied to greater and lesser degree, and this ordering drives a tetragonal distortion. For x from about 0.3 to 0.5, the X-ray patterns show diffuse peaks indicative of similar ordering, but this is not long-range order and no tetragonal distortion results. The lower temperature structures also exhibit out-of-phase tilting of the TiO₆ octahedra, setting in at temperatures varying linearly with composition from 105 K for x=0, to about 650 K at x=2/3.     

Metamagnetism in DyBaCo2O5+x, x≈0.5

The temperature dependence of the paramagnetic susceptibility χ_m(T) taken in 2500 Oe, the resistivity ρ(T), and the thermoelectric power α(T) of DyBaCo₂O_5+x, which has Ba and Dy ordered into alternate (001) planes of an oxygen-deficient perovskite, have revealed a phase segregation in the compositional range 0.3?x<0.5. Orthorhombic DyBaCo₂O_5.51 has, in addition, oxygen vacancies ordered into alternate rows of the DyO_0.51 (001) planes; a cold-pressed polycrystalline sample exhibits a first-order insulator-metal transition at T_IM=320 K, a Curie temperature T_C=300 K, and a broadened metamagnetic transition temperature T_M≈265 K in 2500 Oe. A ferromagnetic M-H hysteresis curve fails to saturate at 5 T, and a minority ferromagnetic phase below T_M has a volume fraction that decreases with decreasing temperature, vanishing below 50 K. Oxygen vacancies in the DyBaCo₂O_5.5 phase suppress the metallic state; interstitial oxygen does not. A thermoelectric power α(T)>0 of DyBaCo₂O_5.51 changing continuously across T_IM is interpreted to manifest a metallic minority phase crossing a percolation threshold; α(T) also provides evidence for a progressive excitation of higher-spin Co(III) with increasing temperature from below 50 K to above T_IM. A previous model of the RBaCo₂O_5.5 phase is extended to account for the Ising spin configuration below T_C, the magnetic order in the presence of higher-spin octahedral-site Co(III), and the α(T) data.     

Crystal Structure and Property of Perovskite-Type Oxides Containing Ion Vacancy

Studies on the role of oxygen vacancy in structural change of nonstoichiometric perovskites and a property of oxygen-deficient perovskite-related K₂NiF₄ compounds are reviewed.The structural changes on which the authors focused are cation ordering and lattice distortion. The relationship between the distortion and oxygen vacancy was investigated by comparing the structures of Sr₂(Sr_1-xM_x)TaO_6-d (M = Ca²⁺ and Nd³⁺) solid solutions. It was found that distortion of a perovskite-type lattice decreased with an increasing amount of oxygen vacancies. In order to investigate the relationship between the cation ordering on octahedral sites and oxygen vacancy, structures of stoichiometric Sr_2-xLa_xCo_1-yTa_1+yO₆ and oxygen-deficient Sr_2-xLa_xMg_1-yTa_1+yO_6-d solid solutions were compared. The authors' work reveals that the cation ordering affects the amount of oxygen vacancies in addition to cation charge and size.     

Long-range magnetic ordering of quasi-one-dimensional S=1/2 Heisenberg antiferromagnet Sr2Cu(PO4)2

Magnetic properties of quasi-one-dimensional S=1/2 Heisenberg antiferromagnet Sr₂Cu(PO₄)₂ were investigated by temperature and field dependence of AC susceptibility down to 0.03 K. A sharp peak was observed at on the temperature dependence of AC susceptibility indicating long-range magnetic ordering. Taking into account the exchange constant, (Hamiltonian =J∑S_iS_i+1), the ratio k_BT_N/J is 0.06%. Sr₂Cu(PO₄)₂ is, therefore, one of the best one-dimensional Heisenberg antiferromagnet known so far.     

ε-optimality criteria for convex programming problems via exact penalty functions

In this paper, we present-optimality criteria for convex programming problems associated with exact penalty functions. Several authors have given various criteria under the assumption that such convex problems and the associated dual problems can be solved. We assume the solvability of neither the convex problem nor the dual problem. To derive our criteria, we estimate the size of the penalty parameter in terms of an-solution for the dual problem.     

A path following algorithm for a class of convex programming problems

We present a primal-dual path following interior algorithm for a class of linearly constrained convex programming problems with non-negative decision variables. We introduce the definition of a Scaled Lipschitz Condition and show that if the objective function satisfies the Scaled Lipschitz Condition then, at each iteration, our algorithm reduces the duality gap by at least a factor of (1–/n), where is positive and depends on the curvature of the objective function, by means of solving a system of linear equations which requires no more than O(n³) arithmetic operations. The class of functions having the Scaled Lipschitz Condition includes linear, convex quadratic and entropy functions.     

Phase separation dynamics in driven diffusive systems

We study phase separation dynamics in a driven diffusive system. Our simulations are based on the Cahn-Hilliard equation with an additional flux term due to an external field. We study the dynamical scaling parallel and perpendicular to the field. A crossover is observed from isotropic domains at early times to extremely anisotropic domains at later times. We find that the inverse interfacial density (an isotropic measure of the domain size) increases ast , with =1/3, from early times independent of the field strength, even though we do not observe dynamical scaling during these times. Our results indicate that a growth exponent =1/3 may be more universal than previously expected. We analyze the dynamics in terms of surface driven instabilities and one-dimensional solitary waves.