The problems of the continuity of homomorphisms between Banach algebras have been studied widely for the last two decades to obtain various fruitful results, yet it is far from characterizing the calss of Banach algebras for which each homomorphism from a member of the class into a Banach algebra is conitnuous. For commutative Banach algebras A and B a simple proof shows that every homomorphism .theta. from A into B is continuous provided that B is semi-simple, however, with a non semi-simple Banach algebra B examples of discontinuous homomorphisms from C(K) into B have been constructed by Dales [6] and Esterle [7]. For non commutative Banach algebras the problems of automatic continuity of homomorphisms seem to be much more difficult. Many positive results and open questions related to this subject may be found in [1], [3], [5] and [8], in particular most recent development can be found in the Lecture Note which contains [1]. It is well-known that a$^{*}$-isomorphism from a $C^{*}$-algebra into another $C^{*}$-algebra is an isometry, and an isomorphism of a Banach algebra into a $C^{*}$-algebra with self-adjoint range is continuous. But a$^{*}$-isomorphism from a $C^{*}$-algebra into an involutive Banach algebra is norm increasing [9], and one can not expect each of such isomorphisms to be continuous. In this note we discuss an isomorphism from a commutative $C^{*}$-algebra into a commutative Banach algebra with dense range via separating space. It is shown that such an isomorphism .theta. : A.rarw.B is conitnuous and maps A onto B is B is semi-simple, discontinuous if B is not semi-simple.

In this paper we shall continue the study of the approximation theorems in the double pseudodifferential operators as in the single pseudodifferential operators.

It is the purpose of this paper to give some remarks on finite fields. We shall show that the little theorem of Fermat, Euler's criterion for quadratic residue mod p, and other few theorems in the number theory can be derived from the theorems in theory of finite field K=GF(p), where p is a prime. The forms of some irreducible ploynomials over K-GF(p) will be given explicitly.

Let X and Y be normed linear spaces. An operator T defined on X with the range in Y is continuous in the sense that if a sequence {x$_{n}$} in X converges to x for the weak topology .sigma.(X.X') then {Tx$_{n}$} converges to Tx for the norm topology in Y. We shall denote the class of such operators by WC(X, Y). For example, if T is a compact operator then T.mem.WC(X, Y). In this note we discuss relationships between WC(X, Y) and the class of weakly of bounded linear operators B(X, Y). In the last section, we will consider some characters for an operator in WC(X, Y).).

In this paper, we show a necessary and sufficient condition for QHC spaces to be S-closed. T. Thomson introduced S-closed spaces in [2]. A topological space X is said to be S-closed if every semi-open cover of X admits a finite subfamily such that the closures of whose members cover the space, where a set A is semi-open if and only if there exists an open set U such that U.contnd.A.contnd.Cl U. A topological space X is quasi-H-closed (denote QHC) if every open cover has a finite subfamily whose closures cover the space. If a topological space X is Hausdorff and QHC, then X is H-closed. It is obvious that every S-closed space is QHC but the converse is not true [2]. In [1], Cameron proved that an extremally disconnected QHC space is S-closed. But S-closed spaces are not necessarily extremally disconnected. Therefore we want to find a necessary and sufficient condition for QHC spaces to be S-closed. A topological space X is said to be semi-locally S-closed if each point of X has a S-closed open neighborhood. Of course, a locally S-closed space is semi-locally S-closed.

Internal heat generation is one of the insidious conditions affecting the quality of an industrial product after it is cast, coated, molded, forged or laminated. Frequently, the product is pressed into service before the exothermic chemical reactions in the generic material has been completed. The heat liberated from this continuing chemical reaction or the residual deformation from the rheological activities in the materials must be adequately removed or prevented, or the product may be discolored, warped, weakened or even "ignited" spontaneously. Numerous instances of premature structural failures, product-recalls, and/or system-malfunctions have been recorded in recent history. The Coulee Dam was poured with pre-chilled concrete just to negate this freakish encore. It is well-known that concrete (a non-isotropic conducting medium), for instance, takes 28 days to develop its full strength. During this period of curing it is conceivable that the processes of internal heat generation, heat conduction and heat dissipation take place simultaneously inside the medium.he medium.

I shall not touch on everything that could be included under this heading. What I shall do is as follows: select three aspects of the teaching of geometry in Korea, and then report the present situations and future prospects of the teaching of geometry in Korea as regards these aspects alone.

지금까지 H. Aebli, A. Fricke, R.W. Copeland, G. Steiner, E. Wittmann, R.R.Skemp, Z.P. Dienes등에 의해 Piaget이론의 수학교육적 연구가 상당한 정도로 이루어져 왔다. 그러나 Centre International D'epistemologie Genetique를 중심으로 한 집단사고와 방대한 연구결과를 집약한 소위 'Piaget이론'은 타에 그 종례를 찾아볼 수 없는 포괄적인 것인 바, 지금까지 이루어진 Piaget이론의 수학교육적 접근은 Piaget이론의 한정된 부분의 단편적인 응용에 불과하며, Piaget의 발생적 수학인식론 및 심리학의 중심원리와 연구결과를 반영한 보다 철저한 연구가 요망되고 있다. 본 고는 그 이론적 기초에 관한 연구의 일환으로 1969년에 출판된 Psychologie et pedagogie에 실린 'La didactique des mathematiques'와 1972년 ICMI의 제2차 수학교육국제회의에 기고한 논문 'Comments on mathematical education'에 나타난 수학교육에 대한 Piaget자신의 견해를 그의 수학인식론의 분석적 고찰을 통해 양세화하고, 그 실제적 구현방안을 제시해 본 것이다.