Internal symmetry groups for perfect elastoplasticity under axial‐torsional loadings

Abstract

In the material modeling of experimental axial‐torsional strain control tests, the hoop and radial strains are always unknown, a priori, and hence can not be viewed as inputs. This greatly complicates constitutive model analyses because the resulting differential equations become highly nonlinear. To tackle this problem, we demonstrate two new formulations. By using the two‐integrating factors idea we derive two Lie type systems in the product space M ¹⁺¹?M ¹⁺¹. The Lie algebra is the direct sum so(1, 1)?so(1, 1), and correspondingly the symmetry group is the direct product SO_o (1, 1) ?SO_o (1, 1). Then, by using the one‐integrating factor idea we convert the nonlinear constitutive equations to a Lie type system X=A(X, t)X with A?sl(2, 1, R), a Lie algebra of the special orthochronous pseudo‐linear group SL(2, 1, R). The underlying space is a cone in the pseudo‐Riemann manifold. Consistent numerical methods are also developed according to these Lie symmetries.     

Transient inter-production scheduling based on Petri nets and constraint programming

In this article, we focus on the transient inter-production scheduling problem between two cyclic productions in the framework of flexible manufacturing systems. This problem is first formulated as a reachability problem in timed Petri nets (TPN), then solved using a methodology based on constraint programming. Our work is based on the controlled executions proposed by Chretienne to model the sequence of transition firing dates. Our methodology is based on a preliminary resolution of the state equation between initial and final states in the underlying non-TPN. Then, we choose a duration T _max corresponding to the maximal duration time of the scheduling. For each solution S of the state equation, we build a controlled execution from the sequence of firings in S. After the propagation of firing date constraints and reachability constraints in the TPN, we use constraint programming to enumerate the set of feasible controlled executions.     

Pole‐assignment of dynamic interval systems

Abstract

A simple sufficient condition which guarantees that all the eigenvalues of an interval matrix will lie inside a circle centered at α with radius r is formulated. A design procedure of robust state feedback controllers for dynamic interval systems is demonstrated by an example.     

Singular Lagrangian systems and variational constrained mechanics on Lie algebroids

The purpose of this article is to describe Lagrangian mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles …). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.     

The Jaynes-Cummings Model with a q Analogue of a Coherent State

Abstract

In this paper we study the time evolution of the atomic inversion of the two-level atom which is coupled to the q analogue of a single mode of the bosonic field. The q field under consideration is supposed to be prepared initially in the q analogue of Glauber's coherent state. We find that q deformation of Heisenberg algebra may correspond to some effective nonlinear interaction of the cavity mode.     

Quantum Phase Angles and su(∞)

Abstract

The polar decomposition of the su(2) algebra leads to unitary phase operators which do not close to an algebra with the number operator. It is shown that phase and number operators can be embedded into a larger su(2j + 1) algebra with trigonometric structure constants. In the contraction limit where we pass from the su(2) to the oscillator algebra, the embedding algebra for the phase operators becomes su(∞). The coherent states realization of the su(∞) algebra and its relation to the q-deformed oscillator algebra is briefly discussed.     

Optimal Legal Firing Sequence of Petri Nets Using Linear Programming

Petri nets (PNs) are a reliable graphical and mathematical modeling tool for the formal modeling and validation of systems (W. Reisig, A Primer in Petri Net Design, Springer-Verlag: Berlin, Heidelberg, 1992). Applications of PNs include discrete event dynamic systems (DEDS) that are recognized as being concurrent, asynchronous, distributed, parallel, and/or nondeterministic. It is also a powerful formal method for the analysis of concurrent, embedded, and distributed finite state systems (K. Varpaaniemi, Series A: Research Reports, No. 26, Helsinki University of Technology, Digital Systems Laboratory, Oct. 1993). The reachability analysis of PNs is strategically significant as it captures the dynamic behavior of the system as well as providing efficient verification of the correctness of the model. Few linear programming (LP)-based methods can be found that address the reachability problem, and some of these are suitable for optimal control problems. However, due to an inherent state explosion they are difficult to implement; other methods run easily into deadlock as they lack appropriate mechanisms to avoid the firing of critical transitions (T. Matsumoto and A. Tarek, in Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996-12, pp. 4459–4468). In this paper an improved and easy to implement method is proposed that combines the Optimality Principle and Linear Programming (OP + LP) techniques to find an Optimal Legal Firing Sequence (OLFS) in PNs. This method can be applied to ordinary PNs with self-loops, avoids deadlocks, and can also be used for general PNs having cycles.     

The Lie-Group Shooting Method for Solving Classical Blasius Flat-Plate Problem

In this paper, we propose a Lie-group shooting method to deal with the classical Blasius flat-plate problem and to find unknown initial conditions. The pivotal point is based on the erection of a one-step Lie group element$\mathbf G(T) and the formation of a generalized mid-point Lie group element$\mathbf G(r). Then, by imposing G(T) = G(r) we can derive some algebraic equations to recover the missing initial conditions. It is the first time that we can apply the Lie-group shooting method to solve the classical Blasius flat-plate problem. Numerical examples are worked out to persuade that the novel approach has better efficiency and accuracy with a fast convergence speed by searching a suitable r ∈(0, 1) with the minimum norm to fit the targets.     

Prediction of the containment of HIV infection by antiretroviral therapy – a variable structure control approach

It is demonstrated that the reachability paradigm from variable structure control theory is a suitable framework to monitor and predict the progression of the human immunodeficiency virus (HIV) infection following initiation of antiretroviral therapy (ART). A manifold is selected which characterises the infection‐free steady‐state. A model of HIV infection together with an associated reachability analysis is used to formulate a dynamical condition for the containment of HIV infection on the manifold. This condition is tested using data from two different HIV clinical trials which contain measurements of the CD4+ T cell count and HIV load in the peripheral blood collected from HIV infected individuals for the six month period following initiation of ART. The biological rates of the model are estimated using the multi‐point identification method and data points collected in the initial period of the trial. Using the parameter estimates and the numerical solutions of the model, the predictions of the reachability analysis are shown to be consistent with the clinical diagnosis at the conclusion of the trial. The methodology captures the dynamical characteristics of eventual successful, failed and marginal outcomes. The findings evidence that the reachability analysis is an appropriate tool to monitor and develop personalised antiretroviral treatment.Inspec keywords: microorganisms, patient treatment, drugs, parameter estimationOther keywords: HIV infection, antiretroviral therapy, variable structure control theory, human immunodeficiency virus, antiretroviral drugs, multipoint identification method, parameter estimation, reachability analysis     

Method of algorithmic transformations with applications to chaotic systems

Transformation of phase space utilizing the Z₂-symmetry of the dynamical system allows to simplify the phase portrait and at the same time introduces a family of the systems sharing common Lie algebra. We demonstrate application of this transformation to continuous time-space dynamical systems, as well as to their discrete representations on a lattice in the phase space.     

A new decentralized stabilization design for large‐scale systems

Abstract

A decentralized stabilization problem for a large‐scale system composed of a number of subsystems is investigated. Using Lyapunov stability and the bounds of the solution of the Lyapunov equation, we derive two main results. The first result (Theorem 1) requires checking the negativity of a matrix containing two free parameters to test the decentralized stabilizability of the whole system. The second result (Theorem 2) determines the ranges of two free parameters to satisfy Theorem 1 such that the decentralized local state feedbacks guarantee the whole large‐scale system is stabilized. The matching condition for each subsystem is not necessary in this paper. The results are also summarized using a flow chart which represents the algorithm for decentralized stabilization.     

Lie symmetries and exact solutions of a new generalized Hirota–Satsuma coupled KdV system with variable coefficients

A new generalized Hirota–Satsuma coupled KdV system with variable coefficients is examined for Lie symmetry group and admissible forms of the coefficients with the help of the symmetry method based on the Fréchet derivative of the differential operators. An optimal system, of non-equivalent (non-conjugate) one dimensional sub-algebras of the symmetry algebra of the KdV system, having ten basic fields is determined. Using the non-equivalent Lie ansätze, for each essential vector field, the nonlinear system is reduced to systems of ordinary differential equations, and some special exact solutions of the KdV system are constructed.     

Integrability and non-integrability in quantum mechanics

The concepts of integrability, non-integrability and chaos in quantum mechanics are examined, and it is indicated that they all are sensibly definable only in connection with the corresponding properties of their classical analogues. The concrete examples concern the quantum and classical properties of dynamic systems on SU₃ algebra and classically integrable but quantum nonintegrable system with two degrees of freedom.     

Using Extended Timed Petri Nets to evaluate and validate timing performance of hard real‐times software

Abstract

An important issue in real‐time systems development is to check the satisfaction of timing requirements. In this paper, we propose a formal model of the Extended Timed Petri Nets (ETPN) to model real‐time software systems. Some basic types of transitions are used to model the structures among software processes. The general equations for evaluating the performance in firing the basic transitions of ETPN are presented. The timing performances of a modelled software system can be evaluated using the corresponding ETPN. The evaluated results are validated with the corresponding timing constraints for checking the satisfaction of the results. The systematic procedure for evalution and validation are also discussed.     

Periodic dynamics of coupled cell networks II: cyclic symmetry

A coupled cell network is a directed graph whose nodes represent dynamical systems and whose directed edges specify how those systems are coupled to each other. The typical dynamic behaviour of a network is strongly constrained by its topology. Especially important constraints arise from global (group) symmetries and local (groupoid) symmetries. The H/K theorem of Buono and Golubitsky characterises the possible spatio-temporal symmetries of time-periodic states of group-equivariant dynamical systems. A version of this theorem for group-symmetric networks has been proved by Josi? and Török. In networks, spatial symmetries correspond to synchrony of cells, and spatio-temporal symmetries correspond to phase relations between cells. Associated with any coupled cell network is a canonical class of admissible ODEs that respect the network topology. A pattern of synchrony or phase relations in a hyperbolic time-periodic state of such an ODE is rigid if the pattern persists under small admissible perturbations. We characterise rigid patterns of synchrony and rigid phase patterns in coupled cell networks, on the assumption that the periodic state is fully oscillatory (no cell is in equilibrium) and the network has a basic property, the rigid phase property. We conjecture that all networks have the rigid phase property, and that in any path-connected network an admissible ODE with a hyperbolic periodic state can always be perturbed to make the perturbed periodic state fully oscillatory. Our main result states that in any path-connected network with the rigid phase property, every rigid pattern of phase relations can be characterised in two stages. First, sets of cells form synchronous clumps according to a balanced equivalence relation. Second, the corresponding quotient network has a cyclic group of automorphisms, and the phase relations are induced by associating a fixed phase shift with a generator of this group. Thus the clumps of synchronous cells form a discrete rotating wave. As a corollary, we prove an analogue of the H/K theorem for any path-connected network. We also discuss the non-path-connected case.     

Mechanical strength of chip scale package with various underfills under accelerated thermal condition

Abstract

The bending strength of chip scale package (CSP) with various underfills was evaluated under thermal shock condition. The maximum bending strength and deflection of package decreased with increasing number of thermal shock cycles. The bending strength of package increased with increasing glass transition temperature T _g and damping property of the underfill, while the bending deflection decreased with increasing T _g and damping property of the underfill.     

Experimental detection of entanglement via witness operators and local measurements

Abstract

In this paper we address the problem of detection of entanglement using only few local measurements when some knowledge about the state is given. The idea is based on an optimized decomposition of witness operators into local operators. We discuss two possible ways of optimizing this local decomposition. We present several analytical results and estimates for optimized detection strategies for NPT states of 2 × 2 and N × M systems, entangled states in 3 qubit systems, and bound entangled states in 3 × 3 and 2 × 4 systems.     

Robust H 8 output feedback control for general nonlinear systems with structured uncertainty

Abstract

In this paper, the robust H ⁸ output feedback control problem for general nonlinear systems with L ₂‐norm‐bounded structured uncertainties is considered. Sufficient conditions for the solvability of robust performance synthesis problems are represented in terms of two Hamilton‐Jacobi inequalities with n independent variables. Based on these conditions, a state space characterization of a robust H ⁸ output feedback controller solving the considered problem is proposed. An example is provided for illustration.     

Midpoint rule for variational integrators on Lie groups

The midpoint rule provides a standard method to obtain symmetric, symplectic, and second‐order accurate variational integrators for mechanical systems whose configuration manifold is the vector space ?ⁿ. In this work, we discuss how to extend this rule to a generic finite‐dimensional Lie group G while retaining the same properties. We show that the function κ_G(g)=exp(½log(g)), g∈G plays a special role in the theory and, for G=SO(3), we give a compact formula to compute it. We also discuss sufficient conditions for the method to conserve momentum maps associated with left (or right) group actions. As an example, the variational integrator obtained from the midpoint rule is applied to simulating rigid body dynamics. The resulting integrator is compared with state‐of‐the‐art symmetric and second‐order accurate integrators for rigid body motion. Copyright © 2009 John Wiley & Sons, Ltd.     

Call for Papers

Abstract

In a previous paper it was shown that the distribution of measured values for a retrodictively optimal simultaneous measurement of position and momentum is always given by the initial state Husimi function. This result is now generalized to retrodictively optimal simultaneous measurements of an arbitrary pair of rotated quadratures ? _{θ 1} and [pcirc] _{θ 2}. It is shown, that given any such measurement, it is possible to find another such measurement, informationally equivalent to the first, for which the axes defined by the two quadratures are perpendicular. It is further shown that the distribution of measured values for such a measurement belongs to the class of generalized Husimi functions most recently discussed by Wünsche and Büzek. The class consists of the subset of Wódkiewicz's operational probability distributions for which the filter reference state is a squeezed vacuum state.