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Design of Experiments is an important branch of Statistics. One of its key problems is to find minimal Fractions of a Full Factorial Design, which identify a Complete Polynomial Model. This paper shows how to use Computer Algebra and Commutative Algebra techniques and results to produce good classes of solutions to the problem. It is known that most of them can be obtained by means of Gröbner bases, hence they generally depend on the term-order chosen; here we show how to use the Distracted Fractions to yield solutions independent of the term-order.
Introduction
Design of Experiments (DoE) is a branch of Statistics, which has a long tradition in the use of algebraic methods (see for example Box et al. 1978 and Collombier 1996). In general all these methods were developed in the case of binary experiments, with coding levels either {0,1} or {-1,1} and some generalizations to the non-binary case were also developed (see Collombier 1996).
More recently some connections were discovered between classical problems in Statistics and the methods of Computer Algebra. For instance in their recent work Pistone and Wynn (1996) address the problem of identifying polynomial models in general designs. In particular they point out the connection between DoE and Gröbner bases.
This article is a brief survey of recent work on Gröbner bases (Buchberger 1965) of toric ideals and their role in integer programming. Toric varieties and ideals are crucial players in the interaction between combinatorics, discrete geometry, commutative algebra and algebraic geometry. For a detailed treatment of this topic see (Sturmfels 1995). Our survey focuses on the application of toric ideals to integer programming, a specific branch of discrete optimization, and for the sake of brevity we leave details to the references that are included.
We study a family of integer programs associated with a fixed matrix A and the corresponding toric ideal IA. In Section 2, we show that reduced Gröbner bases of IA are test sets for these integer programs. In Section 3, we define the universal Gröbner basis of IA and we identify it as a subset of the Graver basis of A. Section 4 deals with the effect of varying the cost function while keeping the matrix A fixed; there we give a self contained construction of the state polytope and Gröbner fan of IA. Moreover, we show that the edge directions of the state polytope are precisely the elements in the universal Gröbner basis of IA. We conclude in Section 5 with a discussion of practical issues that arise while computing Gröbner bases of toric ideals. In particular we discuss algorithms for finding generating sets for IA.
Two operations are presented for a modular approach to the definition of frameworks for rigorous development of software, formally represented as institutions.
The first generalizes models, allowing them to have more structure than the minimum required by their declared signatures, as happens for software modules, which may have local routines that do not appear in their interface.
The second extends sentences, and their interpretation in models, allowing sentences on richer signatures to be used as formulae for poorer ones.
Combining the application of these operations, powerful institutions can be defined, like those for very abstract entities, or for hyper-loose algebraic specifications.
The compatibility of different sequential applications of these operations and properties of the resulting institutions are also studied.
This paper provides an algorithm that, given two finite algebras A and B each of arbitrary finite similarity type, determines whether or not A and B are categorically equivalent. Although the algorithm is not practical in general, we consider some conditions on the algebras that improve its performance.
In recent years, the problem of incorporating a set-building type-constructor into a domain theoretic data model has been addressed by different authors. In Jung and Puhlmann (1995) and Puhlmann (1995) we have shown why the so-called snack powerdomain is particularly suitable for modelling a set constructor. We obtain a generalized database model that covers the nested relational model.
While, with the snack powerconstruction, the data structure of domain theoretic databases seems clear, suitable operations for the data model are still to be defined.
In this paper we start this task by defining the operations nest and unnest for the passage between different nesting-levels of the snack powerconstruction. These functions are shown to form an embedding-projection pair, a property that the corresponding functions of nested relational databases do not have. This demonstrates the usefulness of the domain-theoretic approach for modelling databases: for the first time we have operators for re-grouping nested data that respect the idea of an information ordering.
The use of the snack powerdomain leads to fairly complex formulas. To help the reader, illustrations and pictorial interpretations of formulas are given throughout the paper.
A system to control a database is used for modelling of robotic mechanisms. This brings up the modelling process of robotic mechanisms to a higher level of abstraction and reduces the problem of numerical complexity reduction of the robotic mechanism model to database updating. Structural System Analysis was used to describe the functionality of the system for modelling of robotic mechanisms. The database model is presented by Extended Model Object-Connections, and all the object types for representation of mathematical expressions in the form of calculating graph are described in detail. The complete system is implemented and tested on the example of a robotic mechanism with six degrees of freedom and on the example of anthropomorphic locomotion robotic mechanism.
This paper is intended to be the opening salvo of the workshop, Computing Futures in Engineering Design (Dym, 1997). Thus, I want to take this privileged moment to ask you to think with me about the role of design in engineering. In particular, I want to reflect upon how design is articulated and how design is taught; about the role of design in engineering education and in the practice of engineering; and about the role that could be played locally and, perhaps, nationally by a center devoted to design education. Because I teach here at Harvey Mudd College (HMC), and because most of you are visitors, I will place my remarks in our context by telling you about what we do here and how that doing has shaped my thinking.
At the Forensic Science Service laboratories in Birmingham, U.K., robots are helping to build what is claimed to be the world's first digitised database of DNA samples taken from convicted criminals.
Rapid development and delivery of high-performance, high-quality new products is the key to survival and success for many firms in today's high technology, global marketplace. However, shrinking time to market for complex, high technology products can be extremely difficult and costly. Even the most successful engineering managers have experienced the escalating costs of overtime, coordination and rework that are so common in fast-track, high technology, product development efforts; and they know the quality problems that can result when coordination breaks down in the face of aggressive—often “impossible”—development schedules.
In this paper, we present an experimental evaluation of several link position tracking control algorithms for rigid-link flexible-joint robot manipulators. To study the performance of the controllers, an IMI 2-link direct-drive planar robot manipulator was modified to approximate linear torsional spring couplings from the actuators to the links. Preliminary experimental results seem to indicate that reduced-order, model-based controllers with an actuator feedback loop provide relatively good link position tracking while a full-order, model-based controller offers some further improvement in link position tracking at the expense of increased computation.
Common geared industrial robots call for force control methods with special properties such as good rejection of frictional disturbances, smoothness of corrective motions, and more. A new method is presented which meets these requirements and provides a high control bandwidth. In the manner of hybrid control, directions of a task frame can be selected to be force, impedance or position controlled. A joint-based inner position loop and a superimposed predictive force controller are used. Practical results include data from a robotic grinding facility. Here, the controller proved robustness and good performance under rough conditions.
The Second International Conference in Genetic Programming (GP-97), like the first, was held on the beautiful Stanford University campus in California under the chairmanship of John Koza. More than 350 people from all over the world gathered together for four days to see presentations, posters, tutorials and trade presentations on a range of topics. In addition to GP and related topics in Genetic Algorithms (GAs) and classifier systems, the up and coming fields of evolvable hardware (EHW) and biocomputing (also called DNA computing) were also represented.
The WESTEC exhibition, held on March 24–27, 1997, at the Los Angeles Convention Center attracted 683 exhibitors and nearly 30,000 manufacturing professionals, the biggest ever event. A wide selection of manufacturing equipment and technologies was displayed in an area of over 250,000 square feet, including machine tools of every description, plastics and integrated design and manufacturing . . . . Over 1000 international equipment lines were exhibited by leading international machine tool builders, and the technology categories included machining, milling, grinding and deburring, cutting tools, drilling and tapping, CAD/CAM, EDM, machine control finishing, laser systems, etc . . . .
While there are many project management tools and software packages available, these are not widely used in the design curriculum at colleges and universities. This may reflect some of the differences between conventional projects and design activities. In particular, the open-ended nature of design activities and the need to clarify the client's intention may lead to the conclusion that conventional project management tools are only useful for the most routine activities in the design process. It is suggested that there is a market for a new set of tools for teaching the management of design. These tools should incorporate the most useful of the current management tools and integrate them with some of the requirements of effective design, including support for clarification of objectives, functional analysis, and generation and evaluation of alternatives.
A general method for the computation of the canonical form of three-systems of infinitesimal screws is presented. The method is particularly simple when the three-system has a basis that is simultaneously perpendicular and reciprocal. However, it can also handle the special or degenerate cases. The method and the concurrent results are important from the theoretical point of view because of the obvious connection with the classification of screw-systems. Moreover, the results are also important for applied kinematics after recent applications of the canonical form of screw-systems to the analysis and synthesis of manipulators and manipulator substructures.
The thesis presented here is that the result of engineering is the design, construction, or operation of systems or their subsystems and components and that the teaching of systems must be central to engineering education. It is maintained that current undergraduate engineering curricula do not give the student adequate appreciation of this major intellectual element of their profession. Five proposals for approaches to correct this deficiency are offered: opportunities for clinical practice throughout all the undergraduate years; the use of distributed interactive simulation technology in semester-long projects; courses or course material on the phenomenology and behavior of systems; use of project management tools in engineering clinics; and encouraging engineering faculty to spend some part of their sabbaticals engaged in system design or operation. Issues of implementation are addressed, including the scaling of these ideas to universities that must meet the needs of large numbers of students.