Axiomatic Method of Measure and Integration (V)

In this paper, we define the RS-measure on Rd, (d ≥ 1) by prescribing the complete system of axioms. Then we prove the existence theorem of the RS-measure and determine all the RS-measures. This is a new result

Axiomatic Method of Measure and Integration (XIV)

In this paper, we define the Lebesgue type measure and the Lebesgue type integral on a Riemannian manifold and study their fundamental properties. These results are the new results

Study on the Spectra of Hydrogen Type Ions

In this paper, we study the spectra of hydrogen type ions by using the method of Natural Statistical Physics. As for the results in this paper, we refer to Y. Ito [20] and [44]. Here I express my heartfelt gratitude to my wife Mutuko for her help of typesetting of the TEX-file of this manuscript

Axiomatic Method of Measure and Integration (VII)

In this paper, we define the LS-measure on Rd, (d ≥ 1) by prescribing the complete system of axioms. Then we prove the existence theorem of the LS-measure and we determine all the LS-measures. They are the new results

Axiomatic Method of Measure and Integration (VIII)

In this paper, we define the LS-integral of the LS-measurable functions on Rd, (d ≥ 1). Then we study the fundamental properties of the LS-integral. These facts are the new results

Axiomatic Method of Measure and Integration (III)

In this paper, we define the Lebesgue measure on Rd, (d ≥ 1) by prescribing the complete system of axioms. Then we prove the uniqueness and existence theorem of the Lebesgue measure. This is a new result

Axiomatic Method of Measure and Integration (VI)

In this paper, we define the RS-integral of the RS-measurable functions on Rd, (d ≥ 1). Then we study the fundamental properties of the RS-integral. These facts are the new results

System of Axioms of Euclidean Geometry

In this paper, we define Euclidean geometry and prove the existence theorem of Euclidean geometry by using the axiomatic method. Thereby we give the system of Axioms of Euclidean Geometry and prove its consistency. Thus we give the complete solution of the problem of the foundation of Euclidean geometry

Axiomatic Set Theory

In this paper, we study the new axiomatic set theory. Here we give the new exact definition of the concept of sets and the proof of its existence theorem by using the axiomatic method. Thus we give the complete solution of the problem of axiomatic set theory. Thereby we obtain the complete solution of the problem of the foundation of the mathematics