Abstract:
In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if Z/pZ is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ* = Z/pZ\{0} is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/pZ*. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ* and prove it is cyclic. MML identifier: INT 7, version: 7.8.10 4.99.1005

Abstract:
In this article we continue formalizing probability and randomness started in [13], where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables.

Abstract:
In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The Formalized Mathematics has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.

Abstract:
Metal release from implantable metals and the properties of oxide films formed on alloy surfaces were analyzed, focusing on the highly biocompatible Ti–15Zr–4Nb–4Ta alloy. The thickness and electrical resistance (Rp) of the oxide film on such an alloy were compared with those of other implantable metals. The quantity of metal released during a 1-week immersion test was considerably smaller for the Ti–15Zr–4Nb–4Ta than the Ti–6Al–4V alloy. The potential (E10) indicating a current density of 10 μA cm 2 estimated from the anodic polarization curve was significantly higher for the Ti–15Zr–4Nb–4Ta than the Ti–6Al–4V alloy and other metals. Moreover, the oxide film (4–7 nm thickness) formed on the Ti–15Zr–4Nb–4Ta surface is electrochemically robust. The oxide film mainly consisted of TiO2 with small amounts of ZrO2, Nb2O5 and Ta2O5 that made the film electrochemically stable. The Rp of Ti–15Zr–4Nb–4Ta was higher than that of Ti–6Al–4V, i.e. 0.9 Ω cm2 in 0.9% NaCl and 1.3 Ω cm2 in Eagle's medium. This Rp was approximately five-fold higher than that of stainless steel, which has a history of more than 40 years of clinical use in the human body. Ti–15Zr–4Nb–4Ta is a potential implant material for long-term clinical use. Moreover, E10 and Rp were found to be useful parameters for assessing biological safety.

This paper presents fabrication and testing of a multiple-evaporator and multiple-condenser loop heat pipe (MLHP) with polytetrafluoroethylene (PTFE) porous media as wicks. The MLHP has two evaporators and two condensers in a loop heat pipe in order to adapt to various changes of thermal condition in spacecraft. The PTFE porous media was used as the primary wicks to reduce heat leak from evaporators to compensation chambers. The tests were conducted under an atmospheric condition. In the tests that heat loads are applied to both evaporators, the MLHP was stably operated as with a LHP with a single evaporator and a single condenser. The relation between the sink temperature and the thermal resistance was experimentally evaluated. In the test with the heat load to one evaporator, the heat transfer from the heated evaporator to the unheated evaporator was confirmed. In the heat load switching test, in which the heat load is switched from one evaporator to another evaporator repeatedly, the MLHP could be stably operated. The loop operation with the large temperature difference between the heat sinks was also tested. From this result, the stable operation of the MLHP in the various conditions was demonstrated. It was also found that a flow regulator which prevents the uncondensed vapor from the condensers is required at the inlet of the common liquid line when one condenser has higher temperature and cannot condense the vapor in it.

Abstract:
In this article we introduced the isomorphism mapping between cartesian products of family of linear spaces [4]. Those products had been formalized by two different ways, i.e., the way using the functor [:X, Y:] and ones using the functor "product". By the same way, the isomorphism mapping was defined between Cartesian products of family of linear normed spaces also.

Abstract:
In this article, we formalize a set of points on an elliptic curve over GF(p). Elliptic curve cryptography [10], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.

Abstract:
In this article, we formalize Z-module, that is a module over integer ring. Z-module is necassary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm and cryptographic systems with lattices [11].

Abstract:
In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.