The Mathematical Foundations of Computer Science Laboratory does research related to mathematics of computation and computer-aided teaching of mathematics. This includes research on

1. an interactive email-based teaching system under UNIX;

2. the theoretical foundations of genetic algorithms and genetic programming;

3. applications of genetic algorithms such as optimizing shapes of vehicles in a virtual wind tunnel or optimizing game code; d fuzzy logic systems and generalizations;

4. theory of quantum computers;

5. numerical and other applications of matrix methods.

Professor Schmitt's major research fronts may be described as follows:

1. Teaching System. The Mathematical Foundations of Computer Science Laboratory continuously keeps developing an interactive email-based teaching system under UNIX. This system started as a teaching system for English Composition for Japanese students of The University of Aizu (see [L.M. Schmitt, K.T. Christianson, Pedagogical aspects of a UNIX-based network management system for English instruction, System 26 (1998), 567-589]). This research has been continued and further applications have been developed. These are: (i) introduction of attendance checks and (ii) static/randomized quizzes.

   (i) In today's educational environment where motivation of students is low, the instructor must make an extra effort to have students attend class since only in class can under regular circumstance knowledge be communicated. One necessary inconvenience for students in this regard are attendance checks which are mandated by university regulations. The above mentioned teaching system generates a receipt containing a 10-digit random number which the student obtains in class and returns via email to the computer. The computer verifies the random code and automatically prepares statistics about attendance for the instructor which is used to limit admission to exams.

   (ii) The second new module in regard to the teaching system allows development of quizzes using typesetting with LATEX. Typesetting with LATEX is important for the representation of mathematical content. The quizzes are developed through a special file format which can be best described as a slight LATEX-extension. These contain in separate units the LATEX-source to prepare a task-sheet with questions, the answer to questions (fixed strings of characters) which are used to automatically generate a homework checking program, and special comment which is used to give hints to students about the solution after evaluation. The new module is designed in such a way that students can, based upon a graded computer-generated report including individualized solution-hints, resubmit their work for multiple evaluation. The ensueing amount of machine-generated feedback is impossible to achieve by human work alone.

   The quiz module which has a dedicated development-interface for the instructor has been further generalized to include randomized quizzes such that every student in class obtains a different homework task.

   Publications:

   L.M. Schmitt. An Email-based Learning Management System for Mathematics. In: M. Osano et al (eds.), Proc. HC 2008, 11th International Conference on Humans and Computers, The University of Aizu (2008) L.M. Schmitt. Email-based Mathematics Teaching Tools. In: J. Brine & D. Roy (eds.), Proc. Elearning and Usability Technical Meeting at The University of Aizu: Oct 31st 2009, Electronic Proceedings (2010) L.M. Schmitt. New Yearly Editions of Course Materials (quiz series within the above LMS) for four Undergraduate Courses at The University of Aizu. The University of Aizu (2006-2011)

2. Genetic Algorithms and Genetic Programming Applications. Genetic algorithms and genetic programming are by now well-established optimization techniques. Currently, there are two lines of applied research persued in this direction which both use a genetic algorithm setting with paralled fitness evaluation on the Mathematical Foundations of Computer Science Laboratory's own Linux-PC/Mac network: (i) optimization of vehicle shapes in a virtual wind tunnel in regard to the wind resistance force (with S. Takako, Y. Takemura, H. Tabata) and (ii) optimization (improvement) of a chess program (GNU chess) using a dedicated genetic algorithm (with T. Mitsuta, R. Nitta).

   (i) We have developed and continue to develop a Linux-PC/Mac network on which a parallel genetic algorithm runs which optimizes vehicles shapes (cars, trucks) in a virtual wind tunnel in regard to the wind resistance force. This requires installation and use of specialized software for meshing and solution of the three-dimensional Navier-Stokes equation. This project has thus far included the following activities: generation of a stochastic model for the real-valued genetic algorithm with parallel fitness evealuation along the lines of [L.M. Schmitt. Theory of Genetic Algorithms. Theoretical Computer Science 259 (2001), 1-61]. Optimization of a car shape which gives a realistic impression and reduces the wind resistance force compared to standard triangalar or rectangular shapes considerably. Optimization of the shape of a truck's shield on top of the drivers cabin. Also here, the wind resistance force compared to standard triangalar, parabolic or rectangular shapes is reduced considerably. And the shapes obtained in the latter experiemtn seem to be new.

   (ii) As an application in the field of Artificial Intelligence, it is also attempted to find a better-playing variation of the GNU-chess program using a genetic algorithm where the fitness functions is measured against the original GNU-chess program. This is work in progress.

   Publications:

   S. Takako. Minimizing Wind resistance of Vehicles with a Parallel Genetic Algorithm. Part I: Theory, B.S. Graduation Thesis. The University of Aizu (2010) Y. Takemura. Minimizing Wind resistance of Vehicles with a Parallel Genetic Algorithm. Part II: Experiments, B.S. Graduation Thesis. The University of Aizu (2010) S. Takako, Y. Takemura & L.M. Schmitt. Minimizing Wind resistance of Vehicles with a Parallel Genetic Algorithm. Preprint in preparation. The University of Aizu (2010-2011) T. Mitsuta & L.M. Schmitt. Optimizing the Performance of GNU-Chess with a Parallel Genetic Algorithm. Proc. HC2010 Conf. The University of Aizu (2010)

Professor Asai's major research topics may be described as follows:

1. Matrix Methods and Applications. Matrix-theoretic high-performance algorithm construction with application to the numerical computation of critically important quantities associated with difficult special functions occurring from 3-D wave equations such as regular Coulomb wave functions, Mathieu functions, spheroidal wave functions, Lame functions and ellipsoidal wave functions. The Bessel function of the first kind, an easier function serves as a model case where two types of computational problems were solved in the complex domain as well as real domain: given a complex order Bessel function, compute a given number of zeros closest to the origin to a specified accuracy; and given a complex number find a specified number of orders of Bessel functions closest to the origin, to a specified accuracy. There are no comparable algorithms known elsewhere which give as accurate an error estimate as ours.

2. Construction of Dictionaries. Construction of hyper English word study dictionary: it is available at http://hidic.u-aizu.ac.jp/. This represents a longrange project, aiming at compiling a database for the study of English words for Japanese students. The main feature is that the dictionary is organized to bring out the best fruits of comparative linguistics in the past century: the usefulness of Indo-European roots as presented in the wonderful book by Calvert Watkins, The American Heritage Dictionary of Indo-European Roots, Second Edition, Houghton Mifflin, 2000. Example: The organizing principle of the dictionary will tell you that the following words are best studied as a group: riddle, garble, crime, decree, discern, secret, hypocrisy (they are all derivatives from the same Indo-European root krei-, meaning "to sieve", indicating that a group of seemingly unrelated words come from the same notion "to sieve"). Another example: the usage of the socalled synonyms "anger, rage and wrath" can be clearly explained by tracing them back to their respective Indo-European root: anger to angh-, meaning painful, rage to rebh-, meaning violent, impetuous, and wrath to wreit-, meaning to turn, twist (anger is akin to angst, angina, rage is akin to rabid, rabies, and wrath is akin to wreath, writhe).