Dopant-modulated pair interaction in cuprate superconductors

A comparison of recent experimental STM data with single-impurity and many-impurity Bogoliubov-de Gennes calculations strongly suggests that random out-of-plane dopant atoms in cuprates modulate the pair interaction locally. This type of disorder is crucial to understanding the nanoscale electronic inhomogeneity observed in BSCCO-2212, and can reproduce observed correlations between the positions of impurity atoms and various aspects of the local density of states such as the gap magnitude and the height of the coherence peaks. Our results imply that each dopant atom modulates the pair interaction on a length scale of order one lattice constant.     

Synthesis of 1,4,5,6-tetrahydropyridine derivatives starting from trans-3-chloro-1,3-alkadien-5-ones

A method has been developed for the synthesis of 3-carbethoxy-2-methyl-6-(oxoalkylidene)-1,4,5,6-tetrahydropyridines by reaction of trans-3-chloro-1,3-alkadien-5-ones with ethyl aminocrotonate. It is shown that the corresponding vinylacetylanic ketones are intermediate products of the reactions.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 4, pp. 497–500, April, 1987.     

Simple Singularities for Hamilton-Jacobi Equations with Max-Concave Hamiltonians and Generalized Characteristics

We consider piecewise smooth viscosity solutions to a Hamilton-Jacobi equation, for which the Hamiltonian is the maximum of a finite number of smooth concave functions. We describe the possible types of “basic” singularities, namely jump discontinuities in either the derivative of the solution or the Hamiltonian, which occur across a smooth hypersurface. Each such type of singularity is described in terms of the properties of the Hamiltonian and the classical characteristics in the regions on either side of the singularity. Where appropriate, singular characteristic equations of the types developed by Melikyan are formulated which can be used in constructions.     

Geometry of Optimal Paths around Focal Singular Surfaces in Differential Games

We investigate a special type of singularity in non-smooth solutions of first-order partial differential equations, with emphasis on Isaacs’ equation. This type, called focal manifold, is characterized by the incoming trajectory fields on the two sides and a discontinuous gradient. We provide a complete set of constructive equations under various hypotheses on the singularity, culminating with the case where no a priori hypothesis on its geometry is known, and where the extremal trajectory fields need not be collinear. We show two examples of differential games exhibiting non-collinear fields of extremal trajectories on the focal manifold, one with a transversal approach and one with a tangential approach.     

Synthesis of 5-substituted furans on the basis of the reactions of 1,3-alkadiynes with Β-dicarbonyl compounds

The reaction of 1,3-hexadiyne, 1,3-heptadiyne, and 1,3-octadiyne with acetylacetone and acetoacetic ester in the presence of manganese(III) and copper(II) acetates proceeds with the participation of both triple bonds and leads to substituted 5-alkynyl- and 5-(5-furyl)furans: in the case of 5-hydroxy-5-methyl-1,3-hexadiyne the terminal triple bond participates selectively in the process as a result of the effective protection of the disubstituted triple bond by the dimethylhydroxymethyl group.Communication 148 in the series Reactions of unsaturated compounds; see [1] for Communication 147.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 6, pp. 738–742, June, 1989.     

Andreev states near short-ranged pairing potential impurities

We study Andreev states near atomic scale modulations in the pairing potential in both s- and d-wave superconductors with short coherence lengths. For a moderate reduction of the local gap, the states exist only close to the gap edge. If one allows for local sign changes of the order parameter, however, resonances can occur at energies close to the Fermi level. The local density of states (LDOS) around such pairing potential defects strongly resembles the patterns observed by tunneling measurements around Zn impurities in Bi2Sr2CaCu2O8+x (BSCCO). We discuss how this phase impurity model of the Zn LDOS pattern can be distinguished from other proposals experimentally.     

Singular boundary characteristics of the Hamilton–Jacobi equation

Situations exist in boundary value problems for first order partial differential equations arising in physics (the Hamilton–Jacobi equation), optimal control theory (the Bellman equation) and the theory of differential games (the Isaacs equation) when the value of the required function is not given on a part of the boundary or not at all, or it is not the limit of the (generalized) solution of the problem. Nevertheless, such conditions are required for constructing the solution (by the method of characteristics, for example). It is shown that the required boundary values can be exposed as a specific continuation of the conditions that are known in the boundary submanifolds of the given part of the boundary. This extension of the conditions is accomplished using the characteristic curves starting in a known submanifold of the boundary and running along the boundary. The characteristics are a generalization of the classical characteristics associated with a partial differential equation. They are called singular characteristics, and the theory of these has been developed in a number of the author's papers. After obtaining these “natural” boundary conditions, the solution is constructed using the conventional method of integrating the equations of the classical characteristics. Conditions of the Dirichlet and Neumann type are considered. The technique is illustrated using a numerical example from the theory of differential games containing a number of parameters.