Meteorology is an area where electronic computer has been utilized since an early stage of its invention and remarkable progress has been achieved with the use of it. Numerical atmospheric models, which have played essential roles in the devepolment of meteorology are outlined briefly with their mathematical aspects, performance, application and future prospect.

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Quasicrystals are newly discovered equiliblate systems, showing diffraction patterns of the sharp and densely distributed spots with the crystallographically disallowed symmetry. These materials open a new field of condensed matter physics. Mathematical aspects of quasicrystals are briefly reviewed, including several general methods constructing quasiperiodic systems, generalized crystallography and fractal character of electronic structures.

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Formula manipulation is surveyed from two viewpoints, as a research tool and research theme in mathematical science. From the former viewpoint, current representative systems on formula manipulation are surveyed and how to utilize them effectively in actual research is explained. From the latter viewpoint, algorithms on symbolic determinant, polynomial GCD, polynomial factorization, standard basis of polynomial ideal, approximate sqaure-free decomposition, and approximate factorization are explained elementarily. By this, the author emphasizes that formula manipulation is now being developed rapidly and it is a very interesting and challenging research theme.

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A practical and effective technique called "Fast Automatic Differentiation(FAD)" computes the partial derivatives with respect to all the variables of a function as well as an estimate of the rounding error incurred in the computed value of the function. It has a time complexity at most a (universal) constant times as large as that of evaluating the function alone. In large-scale numerical analysis and optimization, we can use FAD instead of the conventional techniques of which the complexity depends upon the number of variables. Moreover, with this technique, we can define a numerically meaningful "norm" to yield a convergence criterion for an iterative method. In the last decade, several researchers have proposed essentially the same techniques as FAD in different contexts so that there are several formalisms by which to get to FAD, e.g., "computational graph", "matrix multiplication" and "Lagrange multipliers". Thus, FAD will become one of the fundamental techniques in numerical computation.

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