Guy W. Cherry

Integration in Finite Terms with Special Functions: the Error Function

A decision procedure for integrating a class of transcendental elementary functions in terms of elementary functions and error functions is described. The procedure consists of three mutually exclusive cases. In the first two cases a generalised procedure for completing squares is used to limit the error functions which can appear in the integral to a finite number. This reduces the problem to the solution of a differential equation and we use a result of Risch (1969) to solve it. The third case can be reduced to the determination of what we have termed E-decompositions. The result presented here is the key procedure to a more general algorithm which is described fully in Cherry (1983).

Integration in finite terms with special functions: the logarithmic integral

Since R. Risch published an algorithm for calculating symbolic integrals of elementary functions in 1969 (Traps. Amer. Math. Soc., 139 (1969), pp. 167–189), there has been an interest in extending his methods to include nonelementary functions. In this paper, we use the framework of differential algebra to make precise the notion of integration in terms of elementary functions and logarithmic integrals. Basing our work on a recent extension of Liouville’s theorem on integration in finite terms, we then describe a decision procedure for determining if a given element in a transcendental elementary field has an integral which can be written in terms of elementary functions and logarithmic integrals. This algorithm first examines the structure of the integrand in order to limit the logarithmic integrals which could appear in the integral to a finite number. This allows us to write a general expression for the integral and then use techniques similar to those employed by Risch to calculate the undetermined parts.

A Smalltalk system for algebraic manipulation

This paper describes the design of an algebra system Views implemented in Smalltalk. Views contains facilities for dynamic creation and manipulation of computational domains , for viewing these domains as various categories such as groups, rings, or fields, and for expressing algorithms generically at the level of categories. The design of Views has resulted in the addition of some new abstractions to Smalltalk that are quite useful in their own right. Parameterized classes provide a means for run-time creation of new classes that exhibit generally very similar behavior, differing only in minor ways that can be described by different instantiations of certain parameters. Categories allow the abstraction of the common behavior of classes that derives from the class objects and operations satisfying certain laws independently of the implementation of those objects and operations. Views allow the run-time association of classes with categories (and of categories with other categories), facilitating the use of code written for categories with quite different interpretations of operations. Together, categories and views provide an additional mechanism for code sharing that is richer than both single and multiple inheritance. The paper gives algebraic as well as non-algebraic examples of the above-mentioned features.

Spreadsheet computations in computer algebra

In this paper, we analyze the concept of spreadsheet computing in computer algebra. Numerical spreadsheet programs have a unique, intuitively clear semantics. But when the variables used in a spreadsheet program are allowed to assume symbolic expressions as values, the meaning of spreadsheet computations is not defined that simply. We consider several alternative models for what spreadsheet computations could mean in the symbolic context. We describe the implementation of spreadsheet features, based on one of these models, in the visual computer algebra environment MathScribe.

Digital Television Monitoring Systems: Challenges and Opportunities

Todays broadcast facilities are increasing in complexity by an order of magnitude. The relatively straight-forward media formats of the past are being replaced by elaborate packet protocols that combine multiple video, audio and metadata into a digital multiplex. In such a highly computerized system there are many opportunities for error but there are also opportunities for self correction, intelligent report generation, and operator notification. This paper will describe various system monitoring architectures, categorize the types of errors that can be detected, and discuss a number of strategies for recognizing, logging and correcting errors.

High Speed Networks in the Digital TV Studio

The high speed networks that will soon be available should have the bandwidth necessary to handle all of the networking needs of todays studio. With the advent of high definition television (HDTV) the necessary bandwidth will increase more than five-fold. We are interested in what is an appropriate networked-based architecture for the studio. Can we make do with just a single network for all the activities that are included in a studio? Assuming an affirmative answer to the preceding question, will an asynchronous transfer mode (ATM) network be up to the task? Will it meet the real-time needs of the studio? Will an ATM network have adequate bandwidth for the multiplicity of video signals, particularly for HDTV signals? Or, will other networking technology be necessary such as optical networks based on wavelength division multiplexing?