Algebraic Loop Research Articles

QUANTUM LOOP MODULES AND QUANTUM SPIN CHAINS

We construct level-0 modules of the quantum affine algebra [Formula: see text], as the q-deformed version of the Lie algebra loop module construction. We give necessary and sufficient conditions for the modules to be irreducible. We construct the crystal base for some of these modules and find significant differences from the case of highest weight modules. We also consider the role of loop modules in the recent scheme for diagonalizing certain quantum spin chains using their [Formula: see text] symmetry.

Simulation and modelling of continuous systems. A case study approach

Part 1 Modelling and simulation: basic concepts models and modelling modelling and simulation approach modelling and simulation - a case study approach problems. Part 2 Basic methodologies in solving problems with simulation: continuous systems simulation discrete systems simulation problems. Part 3 Basic concepts in continuous system simulation: basic features and classification of simulation systems the concept of digital continuous simulation systems basic features and classification of digital simulations languages the software structure of continuous simulation languages. Part 4 Tools for the simulation of systems with predominantly continuous character: historical overview of computer simulation tools development examples of equation-oriented simulation languages example of a block oriented language the use of general purpose programming languages simulation in computer-aided design packages analog and hybrid simulation systems. Part 5 Further procedures concerning reliable simulation results: numerical integration methods solving the algebraic loop problems. Part 6 Some specific approaches in simulation: optimization of dynamic systems differential and difference equations with boundary conditions partial differential equations problems. Part 7 Influence of software engineering and modern technology on simulation: influence on modern simulation languages influence on the development of simulation systems using special purpose computers. Part 8 Case studies: cascade control of the hydraulic plant distillation column modelling case study approach in drug pharmacokinetics drug pharmacokinetic model development model reference adaptive control simulation of a diesel engine.

Fibrewise Decomposition of Generalized Suspension Spaces and Loop Spaces

We work in the category \mathbf{Top}^B_B of fibrewise pointed topological spaces over B . Let Γ be a co-Hopf space (which need not be co-associative) in \mathbf{Top}^B_B . The Γ_B -suspension space Γ_BX and the Γ_B -loop space Γ^*_BX of a fibrewise pointed space X over B are defined as generalization of the usual suspension space ΣX and the loop space ΩX respectively, Γ_B -suspension spaces and Γ_B -loop spaces have some properties similar to those of the usual suspension spaces and loop spaces. This is an example of Eckmann–Hilton duality. In this paper, decomposition theorems of Γ_B -suspension space Γ_BX and Γ_B -loop space Γ^*_BX are proved. Short exact sequences of homotopy sets involving Γ_B -suspension spaces or Γ_B -loop spaces are obtained in the category of algebraic loops.

Automated formula manipulation supports object-oriented continuous-system modeling

Digital continuous-system simulation languages are discussed. It is demonstrated how sophisticated automated formula manipulation can be used to automatically generate state-space models from an object-oriented description of a physical system. The two major complications, algebraic loops and structural singularities, occur frequently as a consequence of couplings between submodels (objects), and these difficulties can often be dealt with by automated formula manipulation. A software tool, Dymola, in which the various formula manipulation techniques have been implemented, is presented. Dymola is an object-oriented continuous-system modeling language and a model manipulator that can be used to generate models in several simulation languages. >

Delay structure conditions for identifiability of closed loop systems

When identifying a linear system operating under a linear feedback law, specific restrictions have to be imposed on the presence of delays in the plant and/or the controller in order to obtain identifiability. In this paper the delay structure conditions are investigated under which a least squares direct type prediction error identification method is able to consistently estimate the open loop plant. It is shown that the commonly required condition of strictly properness of the loop transfer function can be relaxed to a condition that relates to the absence of algebraic loops in the closed loop plant models.

Causal augmentation of bond graphs with algebraic loops

An algorithm for the causal augmentation of bond graphs with algebraic loops is given. Unlike previous algorithms, this involves bond graph—rather than equation— manipulation. A number of illustrative examples are given.

DESIS-A Modeling and Simulation Package Based on Bond Graphs

Abstract DESIS (DEscription and SImulation Systems), a package which can be used for the modeling and simulation of linear systems described by bond graphs, is presented. DESIS has a graphical and completely user-friendly interface, where the user only has to draw the bond graph of the system on the micro-computer screen while DESIS generates the state and output equations and simulates them. The equations generated are shown in an orderly manner and the simulation results are shown graphically. The state and output equations are generated in parametric and numeric form by a method which is very similar to the manual one. The user communicates with DESIS by the use of menus and in a question-and-answer mode. At each moment, the package analyses the user's response and sends an error or a warning message if it is necessary. The user can observe the bond graph on the screen and a zoom/pan option facilitates the work and visualization of big bond graphs. DESIS can handle derivative causality and algebraic loops as well. It runs in IBM-PC AT or compatibles, is written in FORTRAN 77 and uses G.K.S (Graphical Kernel System) to generate the graphical interface

Loop-theoretic properties of H-spaces

If X is a topological space with base point, then a based map m: X × X → X is called a multiplication if m restricted to each factor of X × X is homotopic to the identity map of X. The pair (X, m) is then called an H-space. If A is a based topological space and (X, m) is an H-space, then the multiplication m induces a binary operation on the set [A, X] of based homotopy classes of maps of A into X. A classical result due to James [6, theorem 1·1 asserts that if A is a CW-complex and (X, m) is an H-space, then the binary operation gives [A, X] the structure of an algebraic loop. That is, [A, X] has a two-sided identity element and if a, b ∈[A, X], then the equations ax = b and ya = b have unique solutions x, y ∈ [A, X]. Thus it is meaningful to consider loop-theoretic properties of H-spaces. In this paper we make a detailed study of the following loop-theoretic notions applied to H-spaces: inversivity, power-associativity, quasi-commutativity and the Moufang property – see Section 2 for the definitions. If an H-space is Moufang, then it has the other three properties. Moreover, an associative loop is Moufang, and so a homotopy-associative H-space has all four properties. Since many of the standard H-spaces are homotopy-associative, we are particularly interested in determining when an H-space, in particular a finite CW H-space, does not have one of these properties.

On Relaxed Delay Structure Conditions In Closed Loop Identifiability Problems

When identifying a linear discrete-time system operating under a linear feedback law, specific restrictions have to be imposed on the presence of delays in the plant and/or the controller in order to arrive at a consistent estimate of the open loop transfer function. In this paper the delay structure conditions are investigated under which a direct prediction error identification method is able to consistently estimate the open loop plant. It is shown that for multivariable systems the commonly used condition of strictly properness of the loop transfer function can be relaxed to a simply verifiable condition that reflects the absence of algebraic loops in the closed loop plant models.

A Method for the Real Time Solution of the Forward Dynamics Problem for Robots Incorporating Friction

A modular and computationally efficient method for solving the forward dynamics problem of robot mechanisms incorporating Coulomb friction is developed. This hybrid approach incorporates both analog and digital components that facilitate real time solutions. Coulomb friction effects associated with both transmissions and bearings are considered. Moreover, the methods accounts for joint flexibility as well as actuator gyroscopic effects. In our approach, the inverse dynamics formulation is used for solving the forward dynamics problem. The positive definiteness property of the inertia matrix of the dynamic equations of motion is exploited by intentionally introducing algebraic loops so that simultaneous algebraic equations are solved without iterations. By resolving these algebraic loops using linear electronics, one avoids the computational burden and time delays associated with purely digital solutions, thus facilitating real time operation. A proof of stability is also presented. The formulation developed here is useful in cases requiring either or both the inverse and forward dynamics solutions typically associated with design and control.

Holomorphic curves in loop groups

It was observed by Atiyah that there is a correspondence between based gauge equivalence classes ofSU n -instantons overS 4 of charged on the one hand, and based holomorphic curves of genus zero inΩSU n of degreed on the other hand. In this paper we study the parameter space of such holomorphic curves which have the additional property that they lie entirely in the subgroupΩ alg SU n of algebraic loops. We describe a cell decomposition of this parameter space, and compute its complex dimension to be (2n−1)d.

A New Simulation Package CSMP-C and its Application to Control System Simulation and Design

This paper is a summary of a new simulation package CSMP-C, which was developed by a group at SIA (Shenyang Institute of Automaiton) to meet the needs of those engineers who are responsible for the development of advanced control systems. While retaining the traditional features of the CSSL specifications, the CSMP-C package contains several new functional blocks which enhance the availability of the package considerably. The new blocks allow the users to conduct complex studies such as parameter optimization or dynamic optimization for nonlinear systems in a convenient way. The support numerical methods, which are essential to the efficiency and accuracy of the performance of the package, include a unique discontinuity handling algorithm and an iteration scheme for treating algebraic loops developed by the authors. The highlights of the package and application examples are outlined in this paper.

Assigning causality in the case of algebraic loops

When, during causality assignment in bond graph modelling, one is faced with a causality indetermination, one is told to assign “arbitrary” causalities. No rule is given to help except of course that of consistency. However, the different consistent causal assignments do not necessary lead to the same algorithm efficiency. Some very simple assignment rules can directly lead to an optimal solution. These rules are particularly useful when causalities are assigned by a computer program.

Using bond graphs in nonlinear simulation problems

A unique feature of bond-graph techniques is that they provide the modeler with a graphical representa tion of the causality (i.e., the input-output rela tionships) in the underlying system. An understand ing of causality makes the modeler aware of algebraic loops and implicit differential equations at an early stage in the solution process. It also clarifies the effects of adding or deleting features of the model. The modeler can use causal information to create simulation programs for nonlinear systems even when some variables cannot readily be expressed in equation form. The resulting program is equivalent to a set of state equations but contains subroutines instead of algebraic expressions. For many complex systems, using vector bond graphs to represent causal interactions among large subsystems can help the modeler write programs using sets of variables rather than individual variables.

Bond Graphs for Distributed System Models Admitting Mixed Causal Inputs

Bond graphs are used for finite mode representations of distributed system dynamics. As long as all inputs to the system are “efforts” in a causal sense, then no formulation problems exist. However, if some of the system inputs are causal “flows”, then differential causality will exist and extremely tedious, often impossible, algebraic loops must be solved to formulate system equations. A procedure is developed which avoids these algebraic problems by including additional modal compliance in the system model without its associated modal inertia. The result of this approach is a finite mode distributed system model, devoid of artificially induced high frequencies, extremely accurate in a chosen frequency range, and capable of interacting with other distributed system models. The procedure is demonstrated through example.

PSI - An Extended, Interactive Block-Oriented Simulation Program

An interactive, block-oriented simulation program is described. This program is enhanced with some facilities that, in general, only can be found in equation-oriented simulation programs, like optimisation, storing and using variables in different simulation runs, solving algebraic loops etc. The interaction is realized by means of a powerfull command language together with a colour TV as graphical output device.

Localization, Algebraic Loops and H-Spaces II

In a previous work [6] it was shown that by imposing certain finiteness conditions on a nilpotent loop certain algebraic results yielded properties about [X, Y] where X is finite CW and Y is an H-Space. In this sequel we further restrict the category of nilpotent loops to a full subcategory called H-loops which still contains all loops of the form [X, Y], We prove that on this category there is a unique and universal P-localization if P ≠ ∅ which corresponds to topological localization. We also show that if the H-loop is a group then the two concepts of localization agree.The first section of this paper is devoted to the definition and basic properties of H-loops. In the second section we develop the localization construction and prove uniqueness. Finally, in the third section we consider the topological and group theoretic situations.

Localization, Algebraic Loops and H-Spaces I

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.

Lagrange’s Equations for Complex Bond Graph Systems

The standard means of imposing causality to extract state equations for bond graph models of physical systems can be inconvenient when algebraic loops and derivative causality in combination with nonlinear constraints are present. This paper presents an alternative means of writing system differential equations using energy and coenergy state functions and Lagrange’s equations. For certain types of systems, particularly mechanical and electromechanical systems, this indirect means of finding state equations turns out to be very convenient. In this technique, causality is used in a new way to establish generalized coordinates and generalized efforts for nonconservative elements. Finally, it is shown that in some cases in which a Lagrangian can be written by inspection for a complex mechanism, a detailed bond graph for this component is unnecessary and yet the equations of the mechanism can be easily coupled to the bond graph equations for the remainder of the system.

Qualitative analysis of interconnected dynamical systems with algebraic loops: Well-posedness and stability

A qualitative analysis of a large class of dynamical systems endowed with algebraic loops is initiated. Such systems, viewed here as interconnected systems, arise naturally in many applications. Qualitative aspects considered include well-posedness (i.e., existence and uniqueness of solutions) as well as stability and instability in the sense of Lyapunov. Whenever appropriate, the analysis is accomplished at different hierarchical levels, namely, at the subsystem level and at the interconnecting structure level. To illustrate the applicability of the results, some specific examples are considered.