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We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^2 (P. Jones, 1990), in R^n (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' beta numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R^n that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space. 

High-temperature poling eliminates light-scattering domain walls in a relaxor ferroelectric. 

Abstract Interface plays a critical role in determining the physical properties and device performance of heterostructures. Traditionally, lattice mismatch, resulting from the different lattice constants of the heterostructure, can induce epitaxial strain. Over past decades, strain engineering has been demonstrated as a useful strategy to manipulate the functionalities of the interface. However, mismatch of crystal symmetry at the interface is relatively less studied due to the difficulty of atomically structural characterization, particularly for the epitaxy of low symmetry correlated materials on the high symmetry substrates. Overlooking those phenomena restrict the understanding of the intrinsic properties of the as‐ determined heterostructure, resulting in some long‐standing debates including the origin of magnetic and ferroelectric dead layers. Here, perovskite LaCoO₃‐SrTiO₃superlattice (SL) is used as a model system to show that the crystal symmetry effect can be isolated by the existing interface strain. Combining the state‐of‐art diffraction and electron microscopy, it is found that the symmetry mismatch of LaCoO₃‐SrTiO₃SL can be tuned by manipulating the SrTiO₃layer thickness to artificially control the magnetic properties. The work suggests that crystal symmetry mismatch can also be designed and engineered to act as an effective strategy to generate functional properties of perovskite oxides.