Narrow Results

To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert–Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.

This short paper outlines the main aims and objectives of the CAPTeaM project and the activities that took place during the CAPTeaM workshop at ICME14 on Wednesday 14 July 2021.

The elastodynamic response of the transformation-toughened ceramics under an instantaneous phase transformation is investigated. Some composite materials, such as Zirconia toughened ceramics, are the remarkable material, which has a high strength, a high elastic modulus, and an improved toughness, etc. Most of the good qualities are common in many ceramic composite materials. These good qualities are made possible through the phase transformation of composite particles. The transformation toughening utilizes the stress-induced phase transformation of particles, which is accompanied by a volumetric expansion. In this paper a phenomenological model is proposed to describe the situation, which involves a dynamic martensitic transformation in a spherical particle of Zirconia embedded in an infinite elastic matrix. Following the ray methods, we clarify the stress-focusing effect caused by the instantaneous phase transformation in a spherical inclusion of Zirconia. It should be noted that the mechanism in the toughening of ceramics in the steady state does not hold in the dynamic state.