Theory Of Connections Research Articles

Least-squares solution of a class of optimal space guidance problems via Theory of Connections

In this paper, we apply a newly developed method to solve boundary value problems for differential equations to solve optimal space guidance problems in a fast and accurate fashion. The method relies on the least-squares solution of differential equations via orthogonal polynomials expansion and constrained expression as derived via Theory of Connection (ToC). The application of the optimal control theory to derive the first order necessary conditions for optimality, yields a Two-Point Boundary Value Problem (TPBVP) that must be solved to find state and costate. Combining orthogonal polynomials expansion and ToC, we solve the TPBVP for a class of optimal guidance problems including energy-optimal landing on planetary bodies and fixed-time optimal intercept for a target-interceptor scenario. The performance analysis in terms of accuracy shows the potential of the proposed methodology as applied to optimal guidance problems.

Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding.

In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.

Geometrical Modeling for Material Flow Management

INTRODUCTION In this paper, we present the theory of connection (ToC) based approach for formulation of mathematical model of material flow processes. The major advantage in using the theory of connection is that, the same set of functions (or procedures) can be used for formulation of mathematical models of many engineering and management processes (Bjorke, 1995; Davidrajuh, 2000). Theory of connection (ToC), previously known as manufacturing system theory, in the form it is presented here, is due to the work of the Scandinavian School of Systems Theory for the past 30 years; for detailed study of ToC, interested readers are referred to Bjorke (1995). The idea behind ToC is to bring geometry and algebra together: first, geometric modeling is used to model physical phenomena, and then a set of algebraic equations are drawn out of the geometric model, so that by using a computer these equations could be solved. The usefulness of ToC is that, different subsystems of different disciplines can be modeled and integrated by performing the same procedure; this is very important for fields like e-commerce as e-commerce involves diverse disciplines like manufacturing, business management, supplier selection, etc. MODELING APPROACH BASED ON ToC First, the concept of system model is introduced; this is done with the help of a simple electrical network known as the inductor-resistor-capacitor (LRC) network. Then, the simple LRC system model is mapped into the geometrical space with the help of properties matrix. System Model A system consists of three fundamental components, such as elements, connections, and sources. The elements carry all the physical or economical properties of the system. Elements are the building blocks of the physical system. For example, in a LRC network, the resistors, capacitors and inductors are the elements; the property of a resistor is its admittance, whereas a machine element's property could be its processing time, ratio between input items and output items, scrap percentage etc. When there is no connection between the elements, the set of isolated elements is called the primitive system. The connections reflect how the elements influence each other and it represents the structure of the system. The set of connected elements is called the connected system. Finally, the sources reflect the influence between the total system and the environment. Sources are the environment's influence on the system; in an electrical circuit, source s could be current or voltage sources; in production planning, demand of products, startup-setup times, costs involved are some of the sources. Geometrical Spaces, Vectors, and Matrices ToC is based on the use of continuous geometrical 3-space or more typically an n-space volume (a Euclidean space [R.sup.n]). A vector (called a contravariant vector) represents a point in the Euclidean space, or primary space, from the origin to the point; A vector in a 3-space is represented by: x = [x.sup.1][[epsilon].sub.1] + [x.sup.2][[epsilon].sub.2] + [x.sup.3][[epsilon].sub.3], where ( [[epsilon].sub.1],[[epsilon].sub.2],[[epsilon].sub.3]) is the basis vector along the three axes. Whereas, a point (called a covariant vector) in the corresponding dual space of the vector space, defines another vector which can be represented by: a = [a.sub.1][[epsilon].sup.1] + [a.sub.2][[epsilon].sup.2] + [a.sub.3][[epsilon].sup.3], Where ([[epsilon].sup.1],[[epsilon].sup.2],[[epsilon].sup.3]) is the basis co-vector along the three axes. A property matrix keeps properties (or characteristics) of primitive elements, whereas a connection matrix represents the connections between the primitive elements. TOOLBOX OF FUNCTIONS The process of mathematical modeling and simulation approach based on ToC is traditionally done by a toolbox of functions implemented in APL (A Programming Language). …

A conceptual model for ethical business decision making under the influence of personal relationships

This paper proposes a conceptual model for ethical business decision making. The purpose of this model is to explore the ethical implications of personal relationship in business exchanges. Firstly, this paper introduces personal relationship in business exchanges. Secondly, three normative theories of ethics that are relevant to the business environment are presented. Thirdly, a literature study on the existing models and frameworks for ethical business decision making is presented. Finally, a new model is presented; this model is developed using an approach based on the Theory of Connection (TOC). The new model supports inclusion of mathematical modules for autonomous ethical business decision making.

Quantum spin dynamics (QSD): VII.Symplectic structures andcontinuum lattice formulations of gauge field theories

Interesting nonlinear functions on the phase spaces ofclassical field theories can never be quantizedimmediately because the basic fields of the theory becomeoperator-valued distributions. Therefore, one is usuallyforced to find a smeared substitute for such a functionwhich corresponds to a regularization. The smearedfunctions define a new symplectic manifold of their ownwhich is easy to quantize. Finally, one must remove theregulator and establish that the final operator, if itexists, has the correct classical limit.In this paper we begin the investigation of these steps fordiffeomorphism-invariant quantum field theories ofconnections. We introduce a (generalized) projectivefamily of symplectic manifolds, coordinatized by thesmeared fields, which is labelled by a pair consisting ofa graph and another graph dual to it. We show that asubset of the corresponding projective limit can beidentified with the symplectic manifold that one startedfrom. Then we illustrate the programme outlined above byapplying it to the Gauss constraint.This paper also complements, as a side result, earlier work by Ashtekar,Corichi and Zapata who observed that certain operators arenon-commuting on certain states, although the Poissonbrackets between the corresponding classical functionsvanish. These authors showed that this is not acontradiction provided that one refrains from a phasespace quantization but rather applies a quantizationbased on the Lie algebra of vector fields on theconfiguration space of the theory. Here we show that onecan provide a phase space quantization, that is,one can find other functions on the classical phase spacewhich give rise to the same operators but whose Poissonalgebra precisely mirrors the quantum commutator algebra.The framework developed here is the classical cornerstone on which thesemiclassical analysis in a new series of papers called`gauge theory coherent states' is based.

Symmetry reduction for quantizeddiffeomorphism-invariant theories of connections

Given a symmetry group acting on a principal fibre bundle, symmetricstates of the quantum theory of a diffeomorphism-invariant theory ofconnections on this fibre bundle are defined. These symmetricstates, equipped with a scalar product derived from theAshtekar-Lewandowski measure for loop quantum gravity, form aHilbert space of their own. Restriction to this Hilbert space yieldsa quantum symmetry reduction procedure within the framework ofspin-network states, the structure of which is analysed in detail. Threeillustrating examples are discussed: reduction of (3+1)- to(2+1)-dimensional quantum gravity, spherically symmetric quantumelectromagnetism and spherically symmetric quantum gravity. In thelatter system the eigenvalues of the area operator applied to thespherically symmetric spin-network states have the form An∝(n(n + 2))1/2, n = 0,1,2,..., giving An∝n for large n.This result clarifies (and reconciles) the relationship between themore complicated spectrum of the general (non-symmetric) areaoperator in loop quantum gravity and the old Bekenstein proposal thatAn∝n.