# Volume 19 (2011): Issue 1 (January 2011)

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

#### Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

Published Online: 18 Jul 2011
Page range: 1 - 9

#### Abstract

In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]).

Open Access

#### Some Properties of p-Groups and Commutative p-Groups

Published Online: 18 Jul 2011
Page range: 11 - 15

#### Abstract

This article describes some properties of p-groups and some properties of commutative p-groups.

Open Access

#### Riemann Integral of Functions from R into Real Normed Space

Published Online: 18 Jul 2011
Page range: 17 - 22

#### Abstract

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. The proof method follows the [16].

Open Access

#### Normal Subgroup of Product of Groups

Published Online: 18 Jul 2011
Page range: 23 - 26

#### Abstract

In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all jI, the group G = ΠiIGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.

Open Access

#### The Mycielskian of a Graph

Published Online: 18 Jul 2011
Page range: 27 - 34

#### Abstract

Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G) > n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation μ(G) called the Mycielskian of a graph G.

We first define the operation μ(G) and then show that ω(μ(G)) = ω(G) and χ(μ(G)) = χ(G) + 1. This is done for arbitrary graph G, see also [10]. Then we define the sequence of graphs Mn each of exponential size in n and give their clique and chromatic numbers.

Open Access

#### Difference and Difference Quotient. Part IV

Published Online: 18 Jul 2011
Page range: 35 - 39

#### Abstract

In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.

Open Access

#### The Definition of Topological Manifolds

Published Online: 18 Jul 2011
Page range: 41 - 44

#### Abstract

This article introduces the definition of n-locally Euclidean topological spaces and topological manifolds [13].

Open Access

#### More on Continuous Functions on Normed Linear Spaces

Published Online: 18 Jul 2011
Page range: 45 - 49

#### Abstract

In this article we formalize the definition and some facts about continuous functions from R into normed linear spaces [14].

Open Access

#### Cartesian Products of Family of Real Linear Spaces

Published Online: 18 Jul 2011
Page range: 51 - 59

#### 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.

Open Access

#### Formalization of Integral Linear Space

Published Online: 18 Jul 2011
Page range: 61 - 64

#### Abstract

In this article, we formalize integral linear spaces, that is a linear space with integer coefficients. Integral linear spaces are necessary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm that outputs short lattice base and cryptographic systems with lattice [8].