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The sine(hyperbolic)-Gordon hierarchy is shown to be the extension of the modified Korteweg-de Vries (MKdV) hierarchy in the integrodifferential algebra extending the standard differential algebra by means of one antiderivative. The characterization by vanishing residues of the MKdV hierarchy yields the same characterization of the sine(hyperbolic)-Gordon hierarchy in the integrodifferential algebra.

Convergence of a second-order shock-capturing scheme for the system of isentropic gas dynamics with $ {L^\infty }$ initial data is established by analyzing the entropy dissipation measures. This scheme is modified from the classical MUSCL scheme to treat the vacuum problem in gas fluids and to capture local entropy near shock waves. Convergence of this scheme for the piston problem is also discussed.

We introduce a unified particle framework which integrates the phase-field method with multi-material simulation to allow modeling of both liquids and solids, as well as phase transitions between them. A simple elastoplastic model is used to capture the behavior of various kinds of solids, including deformable bodies, granular materials, and cohesive soils. States of matter or phases, particularly liquids and solids, are modeled using the non-conservative Allen-Cahn equation. In contrast, materials—made of different substances—are advected by the conservative Cahn-Hilliard equation. The distributions of phases and materials are represented by a phase variable and a concentration variable, respectively, allowing us to represent commonly observed fluid-solid interactions. Our multi-phase, multi-material system is governed by a unified Helmholtz free energy density. This framework provides the first method in computer graphics capable of modeling a continuous interface between phases. It is versatile and can be readily used in many scenarios that are challenging to simulate. Examples are provided to demonstrate the capabilities and effectiveness of this approach.

It is well known that the second-order cone and the circular cone have many analogous properties. In particular, there exists an important distance inequality associated with the second-order cone and the circular cone. The inequality indicates that the distances of arbitrary points to the second-order cone and the circular cone are equivalent, which is crucial in analyzing the tangent cone and normal cone for the circular cone. In this paper, we provide an alternative approach to achieve the aforementioned inequality. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds. Such a clarification is helpful for further study of the relationship between the second-order cone programming problems and the circular cone programming problems.