Abstract. It is shown that determining a function from its averages over all spheres passing through the origin leads to an explicit representation of the even solutions of the Darboux equation in the exterior of the characteristic cones in terms of the hyperboloidal means of the boundary data on the cones.The purpose of this paper is to derive a representation of the even-solutions of the Darboux equation 2?-i d2V/dxf-(n-l)/t dV/dt-d2V/dt2 = 0 in the exterior of the characteristic cones C0 (t= ± \X\, X e Rn, n odd ^3) in terms of its prescribed values on C0-Let f(X) be a continuous function defined in Rn. The spherical means, SM, off is defined as follows:where X is the center of the sphere of radius p. a denotes a unit vector. When P=\X\, we put SM [/; JST, |Jf|] = SM*/ The main part of this paper is concerned with the problem of determining functions from their integrals over a family of spheres passing through the origin. Specifically, if J(X) is a given function in Rn, the main problem is to find f(X) which satisfies (1) J(X) = SM*/ We shall show that inverting (1) is closely related to the representation of the evensolutions of the Darboux equation. The problem of determining functions from their integrals over spheres of a fixed radius was studied by F. John [3]. In this paper we shall make use of the means values of functions defined in Rn over the confocal hyperboloids which have the origin O and a point X=ra, r=\X\^Q, as foci, and the line OX as the axis of rotational symmetry. The

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Abstract. Modern society demands leaders who are trained with competence to not only approach knowledge but also create new knowledge by comprehensively understanding and applying it, and a leader with character and commitment to share one's ideas with others and be able to accept criticisms. In response to these societal changes, universities are increasingly adopting 'small group discussion-based classes with an attempt to develop and strengthen communication skills through reading, writing and speaking. This paper seeks to introduce a case of a math lecture, where discussionbased class was applied to mathematical education, requiring practical problem-solving through an argumentative thought process.

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This study analyses the cross-application system, one of the diversified college admission processes, based on the entrance examination system by the reorganization of the education curriculum. In addition, this study suggests an improvement direction of mathematics education for students who cross-apply by comparing and analysing academic results in Calculus for those students entering universities through the cross-application system.