Time-independent perturbation theory with Lagrange multipliers

We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.

Lagrange-multiplier regularization of eigenproblem for Jx

We solve the eigenproblem of the angular momentum $$J_x$$ J x by directly dealing with the non-diagonal matrix unlike the conventional approach rotating the trivial eigenstates of $$J_z$$ J z . Characteristic matrix is reduced into a tri-diagonal form following Narducci–Orszag rescaling of the eigenvectors. A systematic reduction formalism with recurrence relations for determinants of any dimension greatly simplifies the computation of tri-diagonal matrices. Thus the secular determinant is intrinsically factorized to find the eigenvalues immediately. The reduction formalism is employed to find the adjugate of the characteristic matrix. Improving the recently introduced Lagrange-multiplier regularization, we identify that every column of that adjugate matrix is indeed the eigenvector. It is remarkable that the approach presented in this work is completely new and unique in that any information of $$J_z$$ J z is not required and only algebraic operations are involved. Collapsing of the large amount of determinant calculation with the recurrence relation has a wide variety of applications to other tri-diagonal matrices appearing in various fields. This new formalism should be pedagogically useful for treating the angular momentum problem that is central to quantum mechanics course.

LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS IN ARROWHEAD FORM

This paper deals with different approaches for solving linear systems of the first order differential equations with the system matrix in the symmetric arrowhead form.Some needed algebraic properties of the symmetric arrowhead matrix are proposed.We investigate the form of invariant factors of the arrowhead matrix.Also the entries of the adjugate matrix of the characteristic matrix of the arrowhead matrix are considered. Some reductions techniques for linear systems of differential equations with the system matrix in the arrowhead form are presented.

Improved Impossible Differentials and Zero-Correlation Linear Hulls of New Structure III

Impossible differential cryptanalysis and zero-correlation linear cryptanalysis are two kinds of most effective tools for evaluating the security of block ciphers. In those attacks, the core step is to construct a distinguisher as long as possible. In this paper, we focus on the security of New Structure III, which is a kind of block cipher structure with excellent resistance against differential and linear attacks. While the best previous result can only exploit one-round linear layer P to construct impossible differential and zero-correlation linear distinguishers, we try to exploit more rounds to find longer distinguishers. Combining the Miss-in-the-Middle strategy and the characteristic matrix method proposed at EUROCRYPT 2016, we could construct 23-round impossible differentials and zero-correlation linear hulls when the linear layer P satisfies some restricted conditions. To our knowledge, both of them are 1 round longer than the best previous works concerning the two cryptanalytical methods. Furthermore, to show the effectiveness of our distinguishers, the linear layer of the round function is specified to the permutation matrix of block cipher SKINNY which was proposed at CRYPTO 2016. Our results indicate that New Structure III has weaker resistance against impossible differential and zero-correlation linear attacks, though it possesses good differential and linear properties.

The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs

Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

Polynomial method for the synthesis of regulators for the special case of multichannel objects with one input variable and several output values

Currently, an urgent task in control theory is the synthesis of regulators for objects with a smaller number of input values compared to output ones, such objects are described by matrix transfer functions of a non-square shape. A particular case of a multichannel object with one input variable and two / three / four output variables is considered; the matrix transfer function of such an object has not a square shape, but one column and two / three / four rows. To calculate the controllers, a polynomial synthesis technique is used, which consists in using a polynomial matrix description of a closed-loop control system. A feature of this approach is the ability to write the characteristic matrix of a closed multichannel system through the polynomial matrices of the object and the controller in the form of a matrix Diophantine equation. By solving the Diophantine equation, the desired poles of the matrix characteristic polynomial of the closed system are set. There are many options for solving the Diophantine equation and one of them is to represent the polynomial matrix Diophantine equation as a system of linear algebraic equations in matrix form, where the matrix of the system is the Sylvester matrix. The choice of the order of the polynomial matrix controller and the order of the characteristic matrix is carried out on the basis of the theorem given in the works of Chi-Tsong Chen, which always holds for controlled objects. If the minimum order of the controller is chosen in accordance with this theorem, and the Sylvester matrix has not full rank, then this means that there are more unknown elements in the system of linear algebraic equations than there are equations. In this case, the solution corresponding to the selected basic minor has free parameters, which are the parameters of the regulators. Free parameters of regulators can be set arbitrarily, which is used to set or exclude some zeros in a closed system. Thus, using various examples of objects with a non-square matrix transfer function, a polynomial synthesis technique is illustrated, which allows not only specifying the poles of a closed system, but also some zeros, which is a significant advantage, especially when synthesizing controllers for multichannel objects.

Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine

The finite element discretization of a tower system based on the two-node Euler-Bernoulli beam is carried out by taking the cubic Hermite polynomial as the form function of the beam unit, calculating the structural characteristic matrix of the tower system, and establishing the wind turbine-nacelle-tower multi-degree-of-freedom finite element numerical model. The equation for calculating the aerodynamic load for any nacelle attitude angle is derived. The effect of the flexible tower vibration feedback on the aerodynamic load of the wind turbine is studied. The results show that, when the stiffness of the tower is large, the effect of having tower vibration feedback or not on the aeroelastic load of the wind turbine is small. For the more flexible tower system, wind-induced vibration time-varying feedback will cause larger aeroelastic load variations, especially the top of the tower overturning moment, thus causing a larger impact on the dynamic behavior of the tower downwind and crosswind. As the flexibility of the tower system increases, the interaction between tower vibration and pneumatic load is also gradually increasing. Taking into account the influence of flexible towers on the aeroelastic load of a wind turbine can help predict the pneumatic load of a wind turbine more accurately and improve the efficiency of wind energy utilization on the one hand and analyze the dynamic behavior of the flexible structure of a wind turbine more accurately on the other hand, which is extremely beneficial to the structural optimization of wind turbine.

Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine

Based on the two-node Euler-Bernoulli beam, the tower system is discretized by finite element method, and the cubic Hermite polynomial is taken as the shape function of the beam element, and the structural characteristic matrix of the tower system is calculated, and the wind turbine-nacelle-tower multi-degree of freedom is established Finite element numerical model. The aerodynamic load calculation formula for any nacelle attitude angle is deduced. The influence of the vibration feedback of the flexible tower on the aerodynamic load of the wind turbine is studied. The results show that when the rigidity of the tower is large, the impact of tower vibration feedback on the aeroelastic load of the wind turbine is small. For a tower system with greater flexibility, the time-varying feedback of wind-induced vibration will cause greater aeroelastic load changes, especially the overturning moment of the tower top, which will cause a greater impact on the dynamic behavior of the tower in the downwind and crosswind directions. As the flexibility of the tower system increases, the interaction between tower vibration and aerodynamic load is gradually increasing. Taking the impact of the flexible tower on the aeroelastic load of the wind turbine into account, on the one hand, helps to predict the wind more accurately. The aerodynamic load of the wind turbine improves the efficiency of wind energy utilization. On the other hand, it can more accurately analyze the dynamic behavior of the flexible structure of the wind turbine, which is extremely beneficial to the structural optimization design of the wind turbine.

A Novel Filtering Recommendation Algorithm for User Emergency Information Adoption

Emergency case data resources are widely distributed and heterogeneous. At the same time, the command of emergency field needs the cooperation of multiple departments. Therefore, it is urgent to establish an emergency analysis and mining platform, realize the sharing and collaboration of emergency data resources among multiple departments, and assist emergency command and scheduling. According to the actual situation of the current emergency, a similarity measure method (TCRD) is proposed to solve this problem by adding temporal information to reflect information adoption, which integrates user context information and temporal information. Firstly, the temporal information of historical adoption behavior is expressed as a binary coded characteristic matrix, and then the characteristic matrix is mapped into a feature vector by using restricted Boltzmann machine, and finally added to the similarity measurement formula. The improved TCRD method can measure the similarity more accurately, and further improve the quality of emergency information adoption recommendation results.