Indirect Reconstruction of Structural Responses Based on Transmissibility Concept and Matrix Regularization

A novel method is proposed based on the transmissibility concept and matrix regularization for indirectly measuring the structural responses. The inputs are some measured responses that are obtained via physical sensors. The outputs are the structural responses corresponding to some critical locations where no physical sensors are installed. Firstly, the transmissibility concept is introduced for expressing the relationship between the measured responses and the indirectly measured ones. Herein, a transmissibility matrix is formulated according to the theory of force identification under unknown initial conditions. Then, in order to reduce the size of the transmissibility matrix, structural responses are reshaped in a form of a matrix by using the concept of moving time windows. According to the matrix form of input-output relationship, indirect reconstruction of responses is boiled down to an optimization equation. Since inverse problem may be ill-conditioned, matrix regularization such as F-norm regularization is then recommended for improving the optimization problem. Herein, the penalty function is defined by using a weighted sum of two F-norm values, which correspond to the estimated responses of physical sensors and the ones of the concerned critical locations, respectively. Numerical simulations and experimental studies are finally carried out for verifying the effectiveness and feasibility of the proposed method. Some results show that the proposed method can be applied for indirectly measuring the responses with good robustness.

Matrix regularization of classical Nambu brackets and super p-branes

We present an explicit matrix algebra regularization of the algebra of volume-preserving diffeomorphisms of the n-torus. That is, we approximate the corresponding classical Nambu brackets using $$ \mathfrak{sl}\left({N}^{\left\lceil \frac{n}{2}\right\rceil },\mathrm{\mathbb{C}}\right) $$ sl N n 2 ℂ -matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the N → ∞ limit. We then apply this approximation to the super 4-brane in 9 dimensions and give a regularized action in analogy with the matrix regularization of the supermembrane. This action exhibits a reduced gauge symmetry that we discuss from the viewpoint of L∞-algebras in a slight generalization to the construction of Lie 2-algebras from Bagger-Lambert 3-algebras.

Diffeomorphisms on the fuzzy sphere

Diffeomorphisms can be seen as automorphisms of the algebra of functions. In matrix regularization, functions on a smooth compact manifold are mapped to finite-size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through matrix regularization. For the case of the fuzzy $$S^2$, we construct the matrix regularization in terms of the Berezin–Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $$S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $$S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$$N$ limit) under general diffeomorphisms.