Volume 17 (2009): Issue 2 (January 2009)

In this article, we give some equality and basic theorems about quaternion numbers, and some special operations.

For a complex valued function defined on its domain in complex numbers the differentiability in a single point and on a subset of the domain is presented. The main elements of differential calculus are developed. The algebraic properties of differential complex functions are shown.

This article presents the proof of Kolmogorov's zero-one law in probability theory. The independence of a family of σ-fields is defined and basic theorems on it are given.

In this article we define second-order partial differentiation of real binary functions and discuss the relation of second-order partial derivatives and partial derivatives defined in [17].

In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions. In addition, we proved properties of complex-valued simple functions.

The article starts with definitions of sets of opposite and inverse numbers of a given number membered set. Next, collective addition, subtraction, multiplication and division of two sets are defined. Complex numbers cases and extended real numbers ones are introduced separately and unified for reals. Shortcuts for singletons cases are also defined.

In this article, the principal n-th root of a complex number is defined, the Vieta's formulas for polynomial equations of degree 2, 3 and 4 are formalized. The solution of quadratic equations, the Cardan's solution of cubic equations and the Descartes-Euler solution of quartic equations in terms of their complex coefficients are also presented [5].

This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson's theorem (that n is prime iff n > 1 and (n - 1)! ≅ -1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function Φ, proves that Φ is multiplicative and that Σ_k|n Φ(k) = n.

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.

In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their properties. In addition, we proved properties of complex-valued simple functions.

In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.

In the [16] has been proven that the multiplicative group Z/pZ^* is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group.

Meanwhile, in cryptographic system like RSA, in which security basis depends upon the difficulty of factorization of given numbers into prime factors, it is important to employ integers that are difficult to be factorized into prime factors. If both p and 2p + 1 are prime numbers, we call p as Sophie Germain prime, and 2p + 1 as safe prime. It is known that the product of two safe primes is a composite number that is difficult for some factoring algorithms to factorize into prime factors. In addition, safe primes are also important in cryptography system because of their use in discrete logarithm based techniques like Diffie-Hellman key exchange. If p is a safe prime, the multiplicative group of numbers modulo p has a subgroup of large prime order. However, no definitions have not been established yet with the safe prime and Sophie Germain prime. So it is important to give definitions of the Sophie Germain prime and safe prime.

In this article, we prove finite subgroup of the multiplicative group of a field is a cyclic group, and, further, define the safe prime and Sophie Germain prime, and prove several facts about them. In addition, we define Mersenne number (M_n), and some facts about Mersenne numbers and prime numbers are proven.

The authors have presented some articles about Lebesgue type integration theory. In our previous articles [12, 13, 26], we assumed that some σ-additive measure existed and that a function was measurable on that measure. However the existence of such a measure is not trivial. In general, because the construction of a finite additive measure is comparatively easy, to induce a σ-additive measure a finite additive measure is used. This is known as an E. Hopf's extension theorem of measure [15].

This article introduces labelled state transition systems, where transitions may be labelled by words from a given alphabet. Reduction relations from [4] are used to define transitions between states, acceptance of words, and reachable states. Deterministic transition systems are also defined.

A pseudorandom number generator plays an important role in practice in computer science. For example: computer simulations, cryptology, and so on. A pseudorandom number generator is an algorithm to generate a sequence of numbers that is indistinguishable from the true random number sequence. In this article, we shall formalize the "Uniform Distribution" that is the idealized set of true random number sequences. The basic idea of our formalization is due to [15].

In this article, we define the Riemann Integral of functions from R into Rⁿ, and prove the linearity of this operator. The presented method is based on [21].

In this article we present definitions, basic properties and some examples of even and odd functions [6].

Based on concepts introduced in [14], semiautomata and leftlanguages, automata and right-languages, and langauges accepted by automata are defined. The powerset construction is defined for transition systems, semiautomata and automata. Finally, the equivalence of deterministic and nondeterministic epsilon automata is shown.