Vibrational Modal Frequencies and Shapes of Two-Span Continuous Timber Flooring Systems

Based on classic vibrational bending theory on beams, this paper provides comprehensive analytical formulae for dynamic characteristics of two equal span continuous timber flooring systems, including frequency equations, modal frequencies, and modal shapes. Four practical boundary conditions are considered for end supports, including free, sliding, pinned, and fixed boundaries, and a total of sixteen combinations of flooring systems are created. The deductions of analytical formulae are also expanded to two unequal span continuous flooring systems with pinned end supports, and empirical equations for obtaining the fundamental frequency are proposed. The acquired analytical equations for vibrational characteristics can be applied for practical design of two-span continuous flooring systems. Two practical design examples are provided as well

Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery

This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix $A$ satisfies the RIP condition $\delta_k^A<1/3$, then all $k$-sparse signals $\beta$ can be recovered exactly via the constrained $\ell_1$ minimization based on $y=A\beta$. Similarly, if the linear map $\cal M$ satisfies the RIP condition $\delta_r^{\cal M}<1/3$, then all matrices $X$ of rank at most $r$ can be recovered exactly via the constrained nuclear norm minimization based on $b={\cal M}(X)$. Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.Comment: to appear in Applied and Computational Harmonic Analysis (2012

Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices

This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown that for any given constant $t\ge {4/3}$, in compressed sensing $\delta_{tk}^A < \sqrt{(t-1)/t}$ guarantees the exact recovery of all $k$ sparse signals in the noiseless case through the constrained $\ell_1$ minimization, and similarly in affine rank minimization $\delta_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ ensures the exact reconstruction of all matrices with rank at most $r$ in the noiseless case via the constrained nuclear norm minimization. Moreover, for any $\epsilon>0$, $\delta_{tk}^A<\sqrt{\frac{t-1}{t}}+\epsilon$ is not sufficient to guarantee the exact recovery of all $k$-sparse signals for large $k$. Similar result also holds for matrix recovery. In addition, the conditions $\delta_{tk}^A < \sqrt{(t-1)/t}$ and $\delta_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ are also shown to be sufficient respectively for stable recovery of approximately sparse signals and low-rank matrices in the noisy case.Comment: to appear in IEEE Transactions on Information Theor

Inference for High-dimensional Differential Correlation Matrices

Motivated by differential co-expression analysis in genomics, we consider in this paper estimation and testing of high-dimensional differential correlation matrices. An adaptive thresholding procedure is introduced and theoretical guarantees are given. Minimax rate of convergence is established and the proposed estimator is shown to be adaptively rate-optimal over collections of paired correlation matrices with approximately sparse differences. Simulation results show that the procedure significantly outperforms two other natural methods that are based on separate estimation of the individual correlation matrices. The procedure is also illustrated through an analysis of a breast cancer dataset, which provides evidence at the gene co-expression level that several genes, of which a subset has been previously verified, are associated with the breast cancer. Hypothesis testing on the differential correlation matrices is also considered. A test, which is particularly well suited for testing against sparse alternatives, is introduced. In addition, other related problems, including estimation of a single sparse correlation matrix, estimation of the differential covariance matrices, and estimation of the differential cross-correlation matrices, are also discussed.Comment: Accepted for publication in Journal of Multivariate Analysi