关于对称矩阵随机变换方差的证明

应用正交变换将对称矩阵对角化,基于随机向量正交变换后独立性的不变性及矩阵迹相关性质,给出一个关于对称矩阵经随机变换后方差的证明,并将该结论推广到更一般情形。     

‐symmetry and potentiel well. The simple well case

In this paper, we consider ‐symmetric perturbations of a semi‐classical self‐adjoint operator on the real axis in the case of a simple well potential Schrödinger operator. We assume that the potential is analytic and we show that the eigenvalues are left real by the perturbation.     

Left‐symmetric algebroids

In this paper, we introduce the notion of a left‐symmetric algebroid, which is a generalization of a left‐symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub‐adjacent Lie algebroid. We construct left‐symmetric algebroids from ‐operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left‐symmetric algebroids. Representations of left‐symmetric algebroids are studied in detail. At last, we study deformations of left‐symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology.     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

Solution of the “Classical” Schrödinger equation for a driven symmetric triple well: A comparison with its classical counterpart

The behavior of a driven symmetric triple well potential has been studied by developing an algorithm where the well‐established Bohmian mechanics and time‐dependent Fourier Grid Hamiltonian method are incorporated and the quantum theory of motion (QTM) phase space structures of the particle are constructed, both in “nonclassical” and “classical” limits. Comparison of QTM phase space structures with their classical analogues shows both similarity as well as dissimilarities. The temporal nature and the spatial symmetry of applied perturbation play crucial roles in having similar phase space structures. © 2016 Wiley Periodicals, Inc.     

Neglect of four- and approximation of one-, two-, and three-center two-electron integrals in a symmetrically orthogonalized basis

A new integral approximation for use in molecular electronic structure calculations is proposed as an alternative to the traditional neglect of diatomic differential overlap models. The similarity between the symmetrically orthogonalized and the original basis functions (assumed orthonormal within each atomic set but nonorthogonal between different centers) is used to construct a robust approximation for the two-electron integrals, with the error being quadratic in the deviation between the products of the functions. Invariance properties of this procedure are rigorously proved. Numerical studies on a representative set of molecules at valence-only minimal basis Hartree-Fock level show that the approximation introduces relatively small errors, encouraging its future application in the semiempirical field.     

Shear-strain induced structural relaxation of Cu Σ3 [110](112) symmetric tilt grain boundary: The role of foreign atoms and temperature

Grain boundaries (GBs) relaxation is a promising and effective strategy to improving GB stability or stabilizing nanocrystalline metals. However, previous studies mainly focused on nanocrystalline pure metals and GB behaviors therein, without considering the role of foreign atoms such as impurity or alloying atoms in GB relaxation. In this work, the shear-strain induced structural relaxation of pure Cu Σ3 [110](112) symmetric tilt GBs (STGBs), and the effects of foreign elements (Fe and Ni) and temperature on the GB relaxation were investigated in detail by molecular dynamics method. The results show that shear strain can trigger the structural relaxation of pure, Fe- and Ni-containing Cu GBs by the emission of Shockley partial dislocations from Cu GBs. Both Fe and Ni have impediment effects on the shear-strain induced GB relaxation, though the content of Fe or Ni atom (0.00165 at.%) is quite low in the GB model. The temperature cannot trigger GB relaxation independently within the considered temperature range, but play a positive role in the shear-strain induced structural relaxations of pure, Fe- and Ni-containing Cu Σ3 [110](112) STGBs. Our work might gain new insights into the mechanically induced GB relaxation in nanocrystalline copper and could be beneficial for improving the stability of Cu GBs.