The Intermodulation Coefficient of an Inhomogeneous Superconductor

The high-T_c cuprate superconductors are now believed to be intrinsically inhomogeneous. We develop a theory to describe how this inhomogeneity affects the intermodulation coefficient of such a material. We show that the continuum equations describing intermodulation in a superconducting layer with spatially varying properties are formally equivalent to those describing an inhomogeneous dielectric with a nonzero cubic nonlinearity. Using this formal analogy, we calculate the effect of inhomogeneity on the intermodulation coefficient in a high-T_c material, using several assumptions about the topology of the layer, and some simple analytical approximations to treat the nonlinearity. For some topologies, we find that the intermodulation critical supercurrent density J_{IMD} is actually enhanced compared to a homogeneous medium, thereby possibly leading to more desirable material properties. We discuss this result in light of recent spatial mappings of the superconducting energy gap in BSCCO-2212.Comment: 26 pages, 9 figures, accepted for publication in the Journal of Applied Physic

Least p-Variances Theory

As a result of a rather long-time research started in 2016, this theory whose structure is based on a fixed variable and an algebraic inequality, improves and somehow generalizes the well-known least squares theory. In fact, the fixed variable has a fundamental role in constituting the least p-variances theory. In this sense, some new concepts such as p-covariances with respect to a fixed variable, p-correlation coefficient with respect to a fixed variable and p-uncorrelatedness with respect to a fixed variable are first defined in order to establish least p-variance approximations. Then, we obtain a specific system called p-covariances linear system and apply the p-uncorrelatedness condition on its elements to find a general representation for p-uncorrelated variables. Afterwards, we apply the concept of p-uncorrelatedness for continuous functions particularly for polynomial sequences and find some new sequences such as a generic two-parameter hypergeometric polynomial of 4F3 type that satisfy such a p-uncorrelatedness property. In the sequel, we obtain an upper bound for 1-covariances, an approximation for p-variances, an improvement for the approximate solutions of over-determined systems and an improvement for the Bessel inequality and Parseval identity. Finally, we generalize the notion of least p-variance approximations based on several fixed orthogonal variables.Comment: 85 pages, 1 figure and 1 tabl