We present computable versions of the Fréchet-Riesz Representation Theorem and the Lax-Milgram Theorem. The classical versions of these theorems play important roles in various problems of mathematical analysis, including boundary value problems of elliptic equations. We demonstrate how their computable versions yield computable solutions of the Neumann and Dirichlet boundary value problems for a simple nonsymmetric elliptic differential equation in the one-dimensional case. For the discussion of these elementary boundary value problems, we also provide a computable version of the Theorem of Schauder, which shows that the adjoint of a computably compact operator on Hilbert spaces is computably compact again.

This paper presents in-store customer behavioral model gathered from RFID (RadioFrequency Identification) tags communication data. Although this kind of research has beenmade by various methods such as interviewing or tracking behind customers, Conventionalresearch methods are made by with the existence of customer tracking research, so far. Forcollection of natural customer behavior, we made a customer in-store behavior research withRFID tags in a real retail store. In a conventional store design theory, it has been thought thatincreasing the length of staying time can raise the amount of money per person. Therefore,the store has been designed in the form that goes inside of a shop around. The experimentalresults suggest that there is a correlation between the spent of time and the length of customerwalking path.

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Abstract.It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data X~[ao(Xx -n, x)rx (n), l> 0 , x, r\ e R" , n ^ 0. Here ri(n) is a characteristic vector, and ao(o, x) is a smooth scalar function of compact support. Under the additional requirements that n = 2 or 3 and that üq(o , x) have the vanishing mean with respect to a , it is shown that a genuine solution exists in a time interval independent of A , and that the formal solution is asymptotic to the genuine solution as À -► oo .