Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in $${\mathfrak {sl}}_2$$ sl 2 corresponds to the KdV hierarchy, and the triple $$(f,0,e_\theta )$$ ( f , 0 , e θ ) , where f is the sum of negative simple root vectors and $$e_\theta $$ e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy.

Modeling of rolling force of ultra-heavy plate accounting for gradient temperature

In this paper, the nonlinear specific plastic power of the Mises criterion is integrated analytically to establish the rolling force model of gradient temperature rolling for an ultra-heavy plate by a new method called the root vector decomposition method. Firstly, the sinusoidal velocity field is proposed in terms of the characteristics of metal flow during ultra-heavy plate rolling, which satisfies the kinematically admissible condition. Meanwhile, the characteristics of the temperature distribution along the thickness direction of the plate during the gradient temperature rolling is described mathematically. Based on the velocity field and the temperature distribution expression, the rolling energy functional is obtained by using the root vector decomposition method, and the analytical solution of rolling force is derived according to the variational principle. Through comparison and verification, the rolling force model solved by the root vector decomposition method in this paper is in good agreement with the measured one, and the maximum error of the rolling force is just 10.21%.

On One Method of Studying Spectral Properties of Non-selfadjoint Operators

In this paper, we explore a certain class of Non-selfadjoint operators acting on a complex separable Hilbert space. We consider a perturbation of a nonselfadjoint operator by an operator that is also nonselfadjoint. Our consideration is based on known spectral properties of the real component of a nonselfadjoint compact operator. Using a technique of the sesquilinear forms theory, we establish the compactness property of the resolvent and obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of nonselfadjoint operators. We obtain a classification of nonselfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vector system. Finally, we obtain an asymptotic formula for the eigenvalues.