The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified. We compare the proposed GPT-based shape representation with a moment-based one.

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We propose a conservation-dissipation formalism (CDF) for coarse-grained descriptions of irreversible processes. This formalism is based on a stability criterion for non-equilibrium thermodynamics. The criterion ensures that non-equilibrium states tend to equilibrium in long time. As a systematic methodology, CDF provides a feasible procedure in choosing non-equilibrium state variables and determining their evolution equations. The equations derived in CDF have a unified elegant form. They are globally hyperbolic, allow a convenient definition of weak solutions, and are amenable to existing numerics. More importantly, CDF is a genuinely nonlinear formalism and works for systems far away from equilibrium. With this formalism, we formulate novel thermodynamics theories for heat conduction in rigid bodies and non-isothermal compressible Maxwell fluid flows as two typical examples. In these examples, the non-equilibrium variables are exactly the conjugate variables of the heat fluxes or stress tensors. The new theory generalizes Cattaneo’s law or Maxwell’s law in a regularized and nonlinear fashion.

Under the diffusion scaling and a scaling assumption on the microscopic component, a non-classical fluid dynamic system was derived in\cite {BGLY} that is related to the system of ghost effect derived in\cite {Sone-2} in different settings. This paper aims to justify this limit system for a non-trivial background profile with slab symmetry. The result reveals not only the diffusion phenomena in the temperature and density, but also the flow of higher order in Knudsen number due to the gradient of the temperature. Precisely, we show that the solution to the Boltzmann equation converges to a diffusion wave with decay rates in both Knudsen number and time.

In this note, we study submanifold geometry of the Atiyah--Hitchin manifold, a double cover of the 2-monopole moduli space, which plays an important role in various settings such as the supersymmetric background of string theory. When the manifold is naturally identified as the total space of a line bundle over S^2, the zero section is a distinguished minimal 2-sphere of considerable interest. In particular, there has been a conjecture about the uniqueness of this minimal 2-sphere among all closed minimal 2-surfaces. We show that this minimal 2-sphere satisfies the ``strong stability condition" proposed in our earlier work, and confirm the global uniqueness as a corollary.