From O\mathcal{O}-operators of 3-Dimensional 3-Lie algebras to 3-Pre-Lie algebras

In this paper, we explicitly determine all $\mathcal{O}$-operators with respect to the adjoint representation of 3-dimensional complex 3-Lie algebras. Furthermore, we provide the induced 3-Pre-Lie algebra structures and the corresponding solutions of the 3-Lie classical Yang-Baxter equation in the 6-dimensional 3-Lie algebras $A\ltimes_{\mathrm{ad}^*} A^*$

Manin triples associated to nn-Lie bialgebras

In this paper, we study the Manin triples associated to $n$-Lie bialgebras. We develop the method of double constructions as well as operad matrices to make $n$-Lie bialgebras into Manin triples. Then, the related Manin triples lead to a natural construction of metric $n$-Lie algebras. Moreover, a one-to-one correspondence between the double of $n$-Lie bialgebras and Manin triples of $n$-Lie algebras be established

Effect of Different Polymer Modifiers on the Long-Term Rutting and Cracking Resistance of Asphalt Mixtures

To evaluate the long-term performances of different polymer-modified asphalt mixtures, three modifiers were chosen to modify AC-13 (defined as the asphalt concrete with the aggregate nominal maximum particle size of 13.2 mm); namely, high viscosity modifier (HVM), high modulus modifier (HMM), and anti-rutting agent (ARA). The deformation and cracking resistance of different polymer-modified mixtures were checked at different aging conditions (unaged, short-term aged, and long-term aged for 5, 10, and 15 days respectively). The results of the Hamburg wheel-track test and uniaxial penetration test (UPT) showed that the rutting resistance of all asphalt mixtures changed in a V-shape as the aging progressed. From the unaged stage to the long-term aging stage (5 days), the rutting resistance decreases gradually. While after the long-term aging stage (5 days), the rutting resistance increases gradually. Results from the semicircular bending test (SCB) and the indirect tensile asphalt cracking test (IDEAL-CT) indicated that the cracking resistance of all the mixtures gradually decline with the deepening of the aging degree, indicating that aging weakens the crack resistance of asphalt mixtures. Additionally, test results showed that the rutting resistance of ARA AC-13 (defined as AC-13 containing ARA) is the best, the cracking resistances of ARA AC-13, HMM AC-13 (defined as AC-13 containing HMM) and HVM AC-13 (defined as AC-13 containing HVM) have no significant difference, and different polymer modifiers had different sensitivities to aging due to the polymer content and the type of modifier. The conclusions of this study help to further understand the long-term performance of polymer-modified asphalt mixtures during service life and to help guide the selection of modifiers for mixtures