A New Proof on Sequence of Fuzzy Topographic Topological Mapping

Fuzzy Topological Topographic Mapping (FTTM) is a model for solving neuromagnetic inverse problem. FTTM consists of four components and connected by three algorithms. FTTM version 1 and FTTM version 2 were designed to present 3D view of an unbounded single current and bounded multicurrent source, respectively. In 2008, Suhana proved the conjecture posed by Liau in 2005 such that, if there exists n number of FTTM, then n-n new elements of FTTM will be generated from it. Suhana also developed some new definitions on geometrical features of FTTM, and discovered some interesting algebraic properties. In this paper, new proof on sequence of FTTM will be presented. In the proof, the sequence of FTTM is transformed into a system of differential equation. | Fuzzy topographic topological mapping | Number Theory | Sequence | Differential Equation | ® 2013 Ibnu Sina Institute. All rights reserved. http://dx.doi.org/10.11113/mjfas.v9n4.106


INTRODUCTION
FTTM is a novel method for solving neuromagnetic inverse problem to determine the current source, i.e. epileptic foci.FTTM Version 1 is developed to present a 3-D view of an unbounded single current source [1,2] in one angle observation (upper of a head model).It consists of three algorithms, which link between four components of the model as shown in Figure 1.The four components of FTTM are Magnetic Contour Plane (MC), Base Magnetic Plane (BM), Fuzzy Magnetic Field (FM) and Topographic Magnetic Field (TM) (Figure 1).MC is a magnetic field on a plane above a current source with z = 0.The plane is lowered down to BM, which is a plane of the current source with z = -h.Then the entire BM is fuzzified into a fuzzy environment (FM), where all the magnetic field readings are fuzzified.Finally, a three dimensional presentation of FM is plotted on BM.The final process is defuzzification of the fuzzified data to obtain a 3-D view of the current source (TM).MI is a plane above a current source with z = 0 containing all grey scale readings (0DN-255DN) of magnetic field.The plane is lowered down to BMI, which is a plane of the current source with z = -h.Then the entire base BMI is fuzzified into a fuzzy environment (FMI), where all the gray scale readings are fuzzified.Finally, a three dimensional presentation of FMI is plotted on BMI.The final process is defuzzification of the fuzzified data to obtain a 3-D view of the current source (TMI).
Generally, FTTM can be represented as In order to extract some geometrical features of FTTM, the exact arrangement of sequence of FTTM is presented in Figure 4 [7].
FTTM 1 , FTTM 2 , FTTM 3 , and FTTM 4 are illustrated in Figure 5 respectively.FTTM 1 can be viewed generally as a square and with MC, BM, FM and TM as vertices and the homeomorphism, namely MC ≅ BM, BM ≅ FM, FM ≅ TM and MC ≅ TM, as edges.FTTM 1 has 4 vertices and 4 edges.Generally a cube is a combination of 2 FTTM.FTTM 3 consists of 12 vertices, 24 edges, 15 faces and 3 cubes.FTTM 4 has 16 vertices, 28 edges, 16 faces and 6 cubes.Consequently, some patterns of vertices, edges, faces and cubes emerge from sequences of FTTM as listed in the Table I. (1) Definition 4 Sequence of Edges of FTTM n [6] The sequence of edges for FTTM n which is eFTTM n are given recursively by equation (3) Definition 6 Sequence of cubes of FTTM n [6] The sequence of cubes for FTTM n which is FTTM 2/1 , FTTM 2/2 , FTTM 2/3 ,.., are given recursively by equation Table 1 show the vertices, edges, faces and cubes for some sequences of FTTM [6].From the table and definitions above, several new theorems on sequence of FTTM can be deduced and presented in following section.

A NEW PROOF ON SEQUENCE OF FTTM
Proving the sequence of FTTM can be very tedious as it involves large sequence of numbers.The previous method [8] of proving sequence of FTTM was by proof by construction.However this method required one to develop geometrical features for all sequence of FTTM [9].However, in this section an alternative method of proving sequence of FTTM is presented, namely by method of differential equations [10].

Theorem 1
The Sequence of edges of FTTM n ; eFTTM n can be represented as 4 8 − = n eFTTM n Proof: Recall Definition 4, sequence of edges can be defined as 8 Equation ( 5) can be viewed as a non-homogenous ordinary differential equation as follows with n S is the general solutions 0 and n T is particular solutions such that Equation ( 7) can be solved by finding the related polynomial for equation (5).In this case, the related polynomial for ( 5) is 1 ) ( − = r r P =0 which implies r = 1.Solution for ( 7) is given by Ar S n = .
To get solution for ( 8), let Bn T n = (10) Substituting n T into (8) will gives From ( 9) and (11), solution for ( 6) can be written as From Table I, initial values for

Theorem 2
The Sequence of faces of FTTM n ; fFTTM n can be represented as 4 5 − = n fFTTM n Proof: Recall Definition 5, sequence of edges can be defined as Similarly, equation ( 12) can be viewed as a nonhomogenous ordinary differential equation as follows with n S is the general solutions and n T is particular solutions such that Equation ( 14) can be solved by finding the related polynomial for equation (12).In this case, the related polynomial for (12) is 1 ) ( − = r r P =0 which implies r = 1.Solution for ( 14) is given by Ar U n = .
To get solution for (8), let Substituting n T into (8) will gives 5 From ( 9) and (11), solution for ( 6) can be written as From Table I, initial value for

CONCLUSION
The aim of this paper is to produce new proof on sequence of edges, faces and cubes of FTTM.
Table II, III, IV show sequences of edges, faces, and cubes of FTTM n, , respectively.As a result, the number of sequence of edges, faces, and cubes are exactly the same as defined in [6].

FTTM
Version 2 is developed to present 3-D view of a bounded multi current source [3] in 4 angles of observation (upper, left, right and back of a head model).It consists of three algorithms, which link between four components of the model.The four components are Magnetic Image Plane (MI), Base Magnetic Image Plane (BMI), Fuzzy Magnetic Image Field (FMI) and Topographic Magnetic Image Field (TMI) (Figure 2).

Definition 5
Sequence of Faces of FTTM n[6] | 182 | The sequence of face for FTTM n which is fFTTM n are given recursively

The 2 Bn
Sequence of cubes of FTTM n ; cFTTM n can be represented ) can be viewed as a non-homogenous ordinary differential equation as follows ) can be solved by finding the related polynomial for equation (19).In this case, the related polynomial for r = 1.Solution for (21) is given by Ar W n = .A W n = ∴(23)To get solution for (22), let

Table 1
Vertices, Edges, Faces and Cubes for Sequence of FTTM

Table II
Comparison between sequences of Edges to Definition 4

Table III
Comparison between sequences of Faces to Definition 5

Table IV
Comparison between sequences of Cubes to Definition 6