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A Hopf-algebraic structure on a vector space which has as basis a family of trees is described. Some applications of this structure to combinatorics and to differential operators are surveyed. Some possible future directions for this work are indicated.
It is shown that a particular bilinear model is both quite general and easy to work with. A basic structure theory is developed with the aid of previous results. Some preliminary ideas are discussed together with the system interconnection, the canonical form, questions of controllability, aspects of observability, and equivalent realizations. It is pointed out that in actually determining equivalent realizations for systems and in the classification of systems, the results available in the study of Lie algebras are of fundamental importance.
New results on the realization of finite-dimensional, discrete-time, internally biaffine systems are presented in this paper. The external behavior of such systems is described by multiaffine functions and the state space is constructed via Nerode equivalence relations. We prove that the state space is an affine space. An algorithm which amounts to choosing a frame for the affine space is presented. Our algorithm reduces in the linear and bilinear case to a generalization of algorithms existing in the literature. Explicit existence criteria for span-canonical realizations as well as an affine isomorphism theorem are given.
Linear systems over the quaternions is discussed. The theory of real, semisimple symmetric systems is outlined.
The algebraic theory of thermodynamics developed in a previous paper is extended to include the algebraic structure that arises from the introduction of a physical body into the theory. The extension is based on very general definitions of both the thermodynamic states of a body and subsystems of that body. The algebraic analysis, which includes bodies in nonuniform states, shows that the set of all thermodynamic states of a body has the same algebraic structure as the set of thermodynamic states and that composite systems are induced by the algebraic structure of thermodynamic states. The analysis also justifies a variational treatment of thermodynamic bodies in uniform as well as nonuniform states. The variational calculation includes all conventional methods of calculation as special cases and helps to illuminate the origin and interpretation of the electrochemical potential.
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
Algebraic structures are discussed for control systems that maintain stability in the presence of resonance uncertainties. Dual algebraic operations serve as elementary connections that propagate the stability of inter-stable subsystems. Frequency responses within complex half-planes define different types of inter-stability. Dominance between incompatible types is discussed. Inter-stability produces sufficient but unnecessary stability conditions, except for conservative systems where the conditions become also necessary. Multivariable systems, colocation of actuator and sensor, and virtual colocation are treated. Instead of passivity, inter-stability relates stability to the mapping of poles and zeros by transfer functions and transfer matrices. Inter-stability determines stability on the subsystem level, is less complex even for multivariable systems, adds design flexibility, and relaxes the dynamic data problem of large systems such as space stations.
Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.
Greedy algorithms exploit problem structure and constraints to achieve linear-time performance. Yet there is still no completely satisfactory way of constructing greedy algorithms. For example, the Greedy Algorithm of Edmonds depends upon translating a problem into an algebraic structure called a matroid, but the existence of such a translation can be as hard to determine as the existence of a greedy algorithm itself. An alternative characterization of greedy algorithms is in terms of dominance relations, a well-known algorithmic technique used to prune search spaces. We demonstrate a process by which dominance relations can be methodically derived for a number of greedy algorithms, including activity selection, and prefix-free codes. By incorporating our approach into an existing framework for algorithm synthesis, we demonstrate that it could be the basis for an effective engineering method for greedy algorithms. We also compare our approach with other characterizations of greedy algorithms.
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
The purpose of this paper is to show how the important problems of linear system theory can be solved concisely for a particular class of linear systems, namely block circulant systems, by exploiting the algebraic structure. This type of system arises in lumped approximations to linear partial differential equations. The computation of the transition matrix, the variation of constants formula, observability, controllability, pole allocation, realization theory, stability and quadratic optimal control are discussed. In principle, all questions which are solved here could also be solved by standard methods; the present paper clearly exposes the structure of the solution, and thus permits various savings in computational effort.
The algebraic structure of convolutional codes are reviewed and sequential syndrome decoding is applied to those codes. These concepts are then used to realize by example actual sequential decoding, using the stack algorithm. The Fano metric for use in sequential decoding is modified so that it can be utilized to sequentially find the minimum weight error sequence.
The concept of Cartan-Vessiot filtered Lie algebra/module is defined in terms of previous feedback linearizations constructed through canonical forms of Pfaffian systems associated with nonlinear systems. Nonlinearization/Cartan-Vessiot algebras are regarded as deformations of linear ones. It is shown how certain problems of interest in nonlinear system theory translate over to algebraic problems involving this sort of algebraic structure.
Exterior differential ideals are discussed, and sets of invariant generators presented, for Reimannian, conformal and projective geometries, and for specializations such as Ricci-flat, self-dual and Einstein-Maxwell theories. The Cartan characteristic integers are explicitly calculated, and involutory basis forms found, for each of these (specialized to four dimensions), exposing their algebraic structure and showing how they generate well-posed sets of partial differential equations.
A generalization of the procedure to study shapes and stability of algebraic nuclear models introduced by Gilmore is presented. One calculates the expectation value of the Hamiltonian with respect to the coherent states of the algebraic structure of the system. Then equilibrium configurations of the resulting energy surface, which depends in general on state variables and a set of parameters, are classified through the Catastrophe theory. For one- and two-body interactions in the Hamiltonian of the interacting Boson model-1, the critical points are organized through the Cusp catastrophe. As an example, we apply this Separatrix to describe the energy surfaces associated to the Rutenium and Samarium isotopes.
The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration of points to the grid redistribution scheme. The evaluation of the weighting mesh is accomplished by utilizing the weight function representing the solution variation and the equidistribution law. The selection of the weight function plays a key role in grid adaptation. A new weight function utilizing a properly weighted boolean sum of various flowfield characteristics is defined. The redistribution scheme is developed utilizing Non-Uniform Rational B-Splines (NURBS) representation. The application of NURBS representation results in a well distributed smooth grid by maintaining the fidelity of the geometry associated with boundary curves. Several algebraic methods are applied to smooth and/or nearly orthogonalize the grid lines. An elliptic solver is utilized to smooth the grid lines if there are grid crossings. Various computational examples of practical interest are presented to demonstrate the success of these methods.
A highly structured and compact algebraic representation of grey-level images is expanded. Addition and multiplication are defined for the set of all grey-level images, which can then be described as polynomials of two variables. Utilizing this new algebraic structure, an innovative, efficient edge-detection scheme is devised. A robust method for linear feature extraction is developed by combining the techniques of a Hough transform and a line follower with this new edge detection scheme. The major advantage of this feature extractor is its general, object-independent nature. Target attributes, such as line segment lengths, intersections, angles of intersection, and endpoints are derived by the feature extraction algorithm and employed during model matching. The feature extractor and model matcher are being incorporated into a distributed robot-control system.
Researchers extend their previous research on a highly structured and compact algebraic representation of grey-level images which can be viewed as fuzzy sets. Addition and multiplication are defined for the set of all grey-level images, which can then be described as polynomials of two variables. Utilizing this new algebraic structure, researchers devised an innovative, efficient edge detection scheme. An accurate method for deriving gradient component information from this edge detector is presented. Based upon this new edge detection system researchers developed a robust method for linear feature extraction by combining the techniques of a Hough transform and a line follower. The major advantage of this feature extractor is its general, object-independent nature. Target attributes, such as line segment lengths, intersections, angles of intersection, and endpoints are derived by the feature extraction algorithm and employed during model matching. The algebraic operators are global operations which are easily reconfigured to operate on any size or shape region. This provides a natural platform from which to pursue dynamic scene analysis. A method for optimizing the linear feature extractor which capitalizes on the spatially reconfiguration nature of the edge detector/gradient component operator is discussed.