Impact of Vlasov Foundation Parameters on the Deflection of a Non-uniform Timoshenko Beam Subject to a Moving Load

Subject of the study is a Timoshenko beam with transverse-variable Young’s modulus. Problem of vibrations of the beam resting on an inertial Vlasov foundation and subjected to a moving force is solved analytically. The Timoshenko beam’s eigenproblem is discussed and the physical sense of the additional band of natural vibrations and the corresponding critical frequency is analytically explained. The impact of the foundation and its parameters on the Timoshenko beam’s vibrations forced by moving force is investigated. Dynamic factors relating to beam deflections are analyzed. Two vibration cases are discussed: forced vibrations (when a moving force is applied to the beam) and free vibrations (after the moving force has been left the beam). The damping effect on vibration is taken into account in the problem solution. The results indicate that appropriate selection of the foundation’s parameters allows for the beam deflection’s significant reduction, while the impact of the shear coefficient in the foundation on the reduction is more pronounced than the impact of other factors.

LWR-Based Fully Homomorphic Encryption, Revisited

Very recently, Costache and Smart proposed a fully homomorphic encryption (FHE) scheme based on the Learning with Rounding (LWR) problem, which removes the noise (typically, Gaussian noise) sampling needed in the previous lattices-based FHEs. But their scheme did not work, since the noise of homomorphic multiplication is complicated and large, which leads to failure of decryption. More specifically, they chose LWR instances as a public key and the private key therein as a secret key and then used the tensor product to implement homomorphic multiplication, which resulted in a tangly modulus problem. Recall that there are two moduli in the LWR instances, and then the moduli will tangle together due to the tensor product. Inspired by their work, we built the first workable LWR-based FHE scheme eliminating the tangly modulus problem by cleverly adopting the celebrated approximate eigenvector method proposed by Gentry et al. at Crypto 2013. Roughly speaking, we use a specific matrix multiplication to perform the homomorphic multiplication, hence no tangly modulus problem. Furthermore, we also extend the LWR-based FHE scheme to the multikey setting using the tricks used to construct LWE-based multikey FHE by Mukherjee and Wichs at Eurocrypt 2016. Our LWR-based multikey FHE construction provides an alternative to the existing multikey FHEs and can also be applied to multiparty computation with higher efficiency.

Two questions concerning covering systems

Ever since Erdős introduced the concept of a covering system in 1950, many questions have arisen regarding the existence of certain types of covering systems. Two of the most famous questions regarding covering systems are the odd covering problem and the minimum modulus problem. In the current paper, we ask two questions that are related to these famous questions and provide results for each.