Hamiltonian system approach to distributed spectral decomposition in networks

Existence of a variational principle for PDEs with symmetries and current conservation

Multiple perspectives can be used to explore rich and complex datasets that are widely used in many applications. However, in real-world applications, the multi-view data are often noisy because of various environmental factors. The key challenge of graph-based multi-view clustering is obtaining a consistent clustering structure. Most graph-based methods learn independently in one view how similar the data points in each view are to one another. However, the consistency of information in a multi-view dataset is easily overlooked, causing an unsatisfactory unified graph matrix. To address this problem, this study proposes a novel self-weighted graph-based framework for multi-view clustering (SGMF). This framework integrates these initial similar graph matrices into a unified graph matrix by assigning different weights to the initial graphs of each view. This algorithm’s goal is to effectively learn a unified graph matrix and eliminate noise or irrelevant information from the dataset. This is the first time a generalized multi-view framework based on the L1-norm has been proposed. L1-norm can be used to improve the robustness of the algorithm. Particularly, SGMF can automatically weigh each view’s initial similar graph matrix to solve the noise problem and obtain a consistent data structure. Simultaneously, a rank constraint is imposed on the graph Laplacian matrix of the final unified matrix, which helps to automatically divide the data points into the appropriate number of clusters. The experimental results on synthetic and real datasets verify the effectiveness and superiority of the proposed method over the latest baseline.

The Graph Convolutional Network (GCN) is a universal relation extraction method that can predict relations of entity pairs by capturing sentences’ syntactic features. However, existing GCN methods often use dependency parsing to generate graph matrices and learn syntactic features. The quality of the dependency parsing will directly affect the accuracy of the graph matrix and change the whole GCN’s performance. Because of the influence of noisy words and sentence length in the distant supervised dataset, using dependency parsing on sentences causes errors and leads to unreliable information. Therefore, it is difficult to obtain credible graph matrices and relational features for some special sentences. In this article, we present a Multi-Graph Cooperative Learning model (MGCL), which focuses on extracting the reliable syntactic features of relations by different graphs and harnessing them to improve the representations of sentences. We conduct experiments on a widely used real-world dataset, and the experimental results show that our model achieves the state-of-the-art performance of relation extraction.

Flocking of a team of Lagrangian agents is investigated in this paper. The motion equation of agents with Lagrangian dynamics and its properties are reviewed. The centrifugal and coriolis effects and gravitational terms in the dynamics of agents have been neglected in previous works on flocking problem. However the dynamics of many dynamical systems which would be the agents in practical applications can be assumed as Lagrangian with the above mentioned terms. The proposed flocking scheme is based on a gradient algorithm in which the collective potential is systematically constructed to cause the velocity of the agents to reach a consensus. It is explained that the suggested algorithm satisfies three rules of flocking demonstrated by Reynolds: cohesion, separation, and alignment. For this purpose the Hamiltonian of a team of agents with Lagrangian dynamics is considered and is treated as the Lyapunov like for the Multi-Agent system. Then considering the properties of the Lagrangian systems and also using the algebraic graph theory, it is shown that the derivative of the Hamiltonian is in order that the Reynolds' rules are satisfied and the conditions under which flocking is achieved are derived. The analytical tools provided here are relayed on algebraic graph theory, matrix theory, and control theory. In simulation results, a group of wheeled mobile robots are used to show the effectiveness of the suggested algorithm.

In this paper the density matrices derived from combinatorial Laplacian matrix of graphs is considered. More specifically, the paper places emphasis on the star-relevant graph, which means adding certain edges on peripheral vertices of star graph. Initially, we provide the spectrum of the density matrices corresponding to star-like graph (i.e., adding an edge on star graph) and present that the Von Neumann entropy increases under the graph operation (adding an edge on star graph) and the graph operation cannot be simulated by local operation and classical communication (LOCC). Subsequently, we illustrate the spectrum of density matrices corresponding to star-alike graph (i.e., adding one edge on star-like graph) and exhibit that the Von Neumann entropy increases under the graph operation (adding an edge on star-like graph) and the graph operation cannot be simulated by LOCC. Finally, the spectrum of density matrices corresponding to star-mlike graph (i.e., adding m nonadjacent edges on the peripheral vertices of star graph) is demonstrated and the relation between the graph operation and Von Neumann entropy, LOCC is revealed in this paper.

Undamped and Undriven Oscillators and LC Circuits. The Effect of Damping. Sinusoidally Driven Oscillators and Circuit Loops. Sums of Sinusoidal Forces or EMF's - Fourier Analysis. Integration in the Complex Plane. Evaluation of Certain Fourier Integrals - Causality, Green's Functions. Electrical Networks. Nonlinear Oscillations. Coupled Oscillators without Damping - Lagrange's Equations. Matrices - Rotations - Eigenvalues and Eigenvectors - Normal Coordinates. Some Examples of Normal Coordinates. Finite One-Dimensional Periodic Systems. Difference Equations. Infinite One-Dimensional Periodic Systems - Characteristic Impedance. Continuous Systems. Wave Equation. Lagrangian Density. Hamilton's Principle. Continuous Systems. Applied Forces. Interacting Systems. Other Linear Problems. Nonlinear Waves. Answers to Selected Problems. Index.

A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler--Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge--Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-C-C* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.

Analytical Mechanics. Foundations of Engineering Mechanics Series

A cart inverted pendulum system is one of the most common case to be considered for testing many control algorithms, since it has some challenging problems associated with non linearity, complexity and underactuated system model. In fact, non-linearity behaviour of the inverted pendulum can be observed easily. Different pendulum angle response can be obtained by giving the same velocity in the cart. The cart inverted pendulum can be understood as an under actuated system since the system has a lower number of actuator than the degrees of freedom. One of the most convenient method to model the inverted pendulum system is to use Lagrange's equation. However, at present, many presented inverted pendulum models have been derived using simplified physical model. This simplified model may lead to problems for the implementation of the control algorithm in a real physical inverted pendulum system. In this paper, an inverted pendulum system model is presented, where a mechanical transmission system and a motor models have been included in the derivation of the inverted pendulum model. Hence, the problems for the control implementation in a real system can be minimized. The mathematical model of the inverted pendulum was derived using Lagrange's equation. The determination of the pole zero of the system is discussed. A simple method of pole placement is proposed to stabilise the pendulum at the desired position of the cart. Matlab simulation results show the effectiveness of the proposed method. And yet, this intuitive approach can lead to better understanding of the control behaviours.

Lagrange's equation is a formula in analytical mechanics used to solve problems with physical system dynamics. It allows mathematical modeling to simplify complex mechanical problems by changing the coordinate system, thus providing a deeper understanding of motion. In this research, a literature study was conducted using the Systemic Literature Review (SLR) method from 30 data sources, 24 of which were indexed by Scopus. A total of 11 articles have been reviewed with a focus on the application of Lagrange's equation in various technologies. The review results show that Lagrange multipliers provide a powerful tool for optimizing energy flow within complex smart grids. The benefits extend beyond smart grids. Lagrange's equations are a powerful mathematical tool applicable to various engineering challenges that involve finding optimal solutions under constraints.

This paper discovers that current variational principle and Noether theorem for different physics systems with (in) finite freedom systems have missed the double extremum processes of the general extremum functional that both is deduced by variational principle and is necessarily taken in deducing all the physics laws, but these have not been corrected for over a century since Noether's proposing her famous theorem, which result in the crisis deducing relevant mathematical laws and all physics laws. This paper discovers there is the hidden logic cycle that one assumes Euler-Lagrange equations, and then he finally deduces Euler-Lagrange equations via the equivalent relation in the whole processes in all relevant current references. This paper corrects the current key mistakes that when physics systems choose the variational extreme values, the appearing processes of the physics systems are real physics processes, otherwise, are virtual processes in all current articles, reviews and (text) books. The real physics should be after choosing the variational extreme values of physics systems, the general extremum functional of the physics systems needs to further choose the minimum absolute extremum zero of the general extremum functional, otherwise, the appearing processes of physics systems are still virtual processes. Using the double extremum processes of the general extremum functionals, the crisis and the hidden logic cycle in current variational principle and current Noether theorem are solved. Furthermore, the new mathematical and physical double extremum processes and their new mathematical pictures and physics for (in) finite freedom systems are discovered. This paper gives both general variational principle and general Noether theorem as well as their classical and quantum new physics, which would rewrite all relevant current different branches of science, as key tools of studying and processing them.

An account of the Lagrangian and Hamiltonian dynamics of the pure Yang–Mills system is presented. This framework is applied to the case of (1+1)-dimensional cylindrical spacetime. Hamiltonian dynamics on the space of connections over a circle is often identified with dynamics on the cotangent bundle of the gauge group by means of the holonomy. In support of this procedure we show that the symplectic structure for Hamiltonian dynamics for connections on a circle is identifiable with the natural symplectic structure on the cotangent bundle of the gauge group.

Some new discretizations of the Euler–Lagrange equation

Plane computer graphics are basic information carriers in many industrial scenarios, such as engineering simulation, automatic control, and software design. Plane computer graphics are generally a kind of digital signals guided by mathematical symbols, and each vertex of a plane computer graph forms a graph matrix. Therefore, linear matrix transformation serves as the most common algorithmic unit to realize various information processing operations. To improve ease of graph matrix computing in practical engineering scenarios, this paper proposes a theoretical scientific programming framework for application of linear matrix transformation in plane computer graphics. Firstly, theoretical basis of linear matrix transformation in homogeneous plane coordinates is displayed and analyzed. Then, the universal theorem about linear transformation of graph matrices is deduced, and corresponding proofs are also given. Finally, a case study is set up to demonstrate the main workflow of the proposed theoretical scientific programming framework. The simulative results reveal feasibility of the proposal.

The Lagrangian description of a system is analysed from a geometric viewpoint in order to find a concept for equivalence of singular Lagrangians generalising that of the regular case. Geometric and gauge equivalence of singular Lagrangians are studied and the authors also give some conditions in which second-order differential equations exist satisfying the dynamical equation on the final constraint submanifold.