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Summary.In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. PreliminariesLet X be a real normed space, let A be a closed-interval subset of R, let f be a function from A into the carrier of X, and let D be a Division of A. A finite sequence of elements of X is said to be a middle volume of f and D if it satisfies the conditions (Def. 1).(Def. 1)(i) len it = len D, and (ii) for every natural number i such that i ∈ dom D there exists a point c of X such that c ∈ rng(f divset(D, i)) and it(i) = vol(divset (D, i)) · c. 17

Microfluidic devices employing dielectrophoresis (DEP) have been widely studied and applied in the manipulation and analysis of single cells. However, several pre-processing steps, such as the preparation of purified target samples and buffer exchanges, are necessary to utilize DEP forces for suspended cell samples. In this paper, a sequential cell-processing device, which is composed of pre-processing modules that employ deterministic lateral displacement (DLD) and a single-cell trapping device employing an electroactive microwell array (EMA), is proposed to perform the medium exchange followed by arraying single cells sequentially using DEP. Two original microfluidic devices were efficiently integrated by using the interconnecting substrate containing rubber gaskets that tightly connect the inlet and outlet of each device. Prostate cancer cells (PC3) suspended in phosphate-buffered saline buffer mixed with microbeads were separated and then resuspended into the DEP buffer in the integrated system. Thereafter, purified PC3 cells were trapped in a microwell array by using the positive DEP force. The achieved separation and trapping efficiencies exceeded 94% and 93%, respectively, when using the integrated processing system. This study demonstrates an integrated microfluidic device by processing suspended cell samples, without the requirement of complex preparation steps.

Summary. In this article, we define the product space of real linear spaces and real normed spaces. We also describe properties of these spaces. The Product Space of Real Linear SpacesThe following propositions are true: (1) Let s, t be sequences of real numbers and g be a real number. Suppose that for every element n of N holds t(n) = |s(n) − g|. Then s is convergent and lim s = g if and only if t is convergent and lim t = 0. (2) Let x, y be finite sequences of elements of R. Suppose len x = len y and for every element i of N such that i ∈ Seg len x holds 0 ≤ x(i) and x(i) ≤ y(i). Then |x| ≤ |y|. (3) Let F be a finite sequence of elements of R. If for every element i of N such that i ∈ dom F holds F (i) = 0, then F = 0. 81