# Volume 20 (2012): Issue 4 (December 2012)

Journal Details
Format
Journal
eISSN
1898-9934
ISSN
1426-2630
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English

0 Articles
Open Access

#### Posterior Probability on Finite Set

Published Online: 02 Feb 2013
Page range: 257 - 263

#### Abstract

In [14] we formalized probability and probability distribution on a finite sample space. In this article first we propose a formalization of the class of finite sample spaces whose element’s probability distributions are equivalent with each other. Next, we formalize the probability measure of the class of sample spaces we have formalized above. Finally, we formalize the sampling and posterior probability.

Open Access

#### Basic Properties of Primitive Root and Order Function

Published Online: 02 Feb 2013
Page range: 265 - 269

#### Abstract

In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].

Open Access

#### Banach’s Continuous Inverse Theorem and Closed Graph Theorem

Published Online: 02 Feb 2013
Page range: 271 - 274

#### Abstract

In this article we formalize one of the most important theorems of linear operator theory - the Closed Graph Theorem commonly used in a standard text book such as [10] in Chapter 24.3. It states that a surjective closed linear operator between Banach spaces is bounded.

Open Access

#### Free ℤ-module

Published Online: 02 Feb 2013
Page range: 275 - 280

#### Abstract

In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.

Open Access

#### Cayley-Dickson Construction

Published Online: 02 Feb 2013
Page range: 281 - 290

#### Abstract

Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers, quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties.

Open Access

#### Contracting Mapping on Normed Linear Space

Published Online: 02 Feb 2013
Page range: 291 - 301

#### Abstract

In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

Open Access

#### Products in Categories without Uniqueness of cod and dom

Published Online: 02 Feb 2013
Page range: 303 - 307

#### Abstract

The paper introduces Cartesian products in categories without uniqueness of cod and dom. It is proven that set-theoretical product is the product in the category Ens [7].

Open Access

#### Program Algebra over an Algebra

Published Online: 02 Feb 2013
Page range: 309 - 341

#### Abstract

We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The execution of a program is changing state according to successor function using supporting terms.

Open Access

#### Isomorphisms of Direct Products of Finite Cyclic Groups

Published Online: 02 Feb 2013
Page range: 343 - 347

#### Abstract

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

Open Access

#### On L1 Space Formed by Complex-Valued Partial Functions

Published Online: 02 Feb 2013
Page range: 349 - 357