MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) vol. 1, 2009

2009·1.1 MB·English

by  Mao, Linfan.

ISBN 1-59973-086-3
VOLUME 1, 2009
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES
April, 2009
Vol.1, 2009 ISBN 1-59973-086-3
Mathematical Combinatorics
(International Book Series)
Edited By Linfan MAO
The Madis of Chinese Academy of Sciences
April, 2009
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Editorial Board
M.Khoshnevisan
School of Accounting and Finance,
Griffith University, Australia
Editor-in-Chief
Linfan MAO Xueliang Li
Chinese AcademyofMathematicsandSystem Nankai University, P.R.China
Email: [email protected]
Science, P.R.China
Email: [email protected] Han Ren
East China Normal University, P.R.China
Editors Email: [email protected]
W.B.Vasantha Kandasamy
S.Bhattacharya Indian Institute of Technology, India
Email: [email protected]
Alaska Pacific University, USA
Email: sbhattacharya@alaskapacific.edu Mingyao Xu
Peking University, P.R.China
An Chang
Email: [email protected]
Fuzhou University, P.R.China
Email: [email protected] Guiying Yan
Chinese AcademyofMathematicsandSystem
Junliang Cai
Science, P.R.China
Beijing Normal University, P.R.China
Email: [email protected]
Email: [email protected]
Y. Zhang
Yanxun Chang
Department of Computer Science
Beijing Jiaotong University, P.R.China
Georgia State University, Atlanta, USA
Email: [email protected]
Shaofei Du
Capital Normal University, P.R.China
Email: [email protected]
Florentin Popescu and Marian Popescu
University of Craiova
Craiova, Romania
Xiaodong Hu
Chinese AcademyofMathematicsandSystem
Science, P.R.China
Email: [email protected]
Yuanqiu Huang
Hunan Normal University, P.R.China
Email: [email protected]
H.Iseri
Mansfield University, USA
Email: hiseri@mnsfld.edu
ii InternationalJournalofMathematicalCombinatorics
No object is mysterious. The mystery is your eye.
By Elizabeth, a British female writer.
Math.Combin.Book.Ser. Vol.1 (2009), 01-08
Study of the Problems of
Persons with Disability (PWD) Using FRMs
W.B. Vasantha Kandasamy
(DepartmentofMathematics,IndianInstituteofTechnology,Chennai-600036)
E-mail: [email protected]
A. Praveen Prakash
(DepartmentofMathematics,GuruNanakCollege,Chennai-600042)
K. Thirusangu
(DepartmentofMathematics,AnnaUniversity,Chennai-600025)
Abstract: Inthispaperwe findtheinterrelations andthehiddenpatternof theproblems
faced by the PWDs and their caretakers using Fuzzy Relational Maps (FRMs). Here we
have taken the problems faced by the rural persons with disabilities in Melmalayanur and
Kurinjipadi Blocks, Tamil Nadu, India. This paper is organized with the following four
sections. Section one is introductory in nature giving the overall contents from the survey
made about PWDs in the above said Blocks. Section two gives description of FRM models
and the attributes taken for the study related with the PWDs and the caretakers, the
FRM model formed using these attributes and their analysis. The third section gives the
suggestions and conclusions derived from thesurvey as well as the FRMmodel.
Key Words: FRM model, fixedpoint, hidden pattern,relational matrix, limit cycle.
AMS(2000): 04A72.
§1. Introduction
A study was conducted taking 93 village panchayats from the Kurinjipadi and Melmalayanur
Blocks. The data reveals only 1.64 percent of the population are PWDs. The male population
is comparatively higher. (60% males and 40% females). 51% are orthopedic followed by 16%
with speech and hearing impaired. Also it is observedfrom the data that 60% are not married
in the reproductive age group; however 73% are found married in the non reproductive age
group. It is still unfortunate to see among the 3508 PWDs in the age group 4 yrs and above
59% of them have not even entered school. Further in the age group 4 to 14, 37% are yet to
be enrolled in the school. Thus the education among the PWDs is questionably poor. Their
living conditions are poor with no proper toilet facilities who are under nourished.
We use FRMs to study the problem taking the attributes of the domain space as the
problems faced by the PWD and the range attributes are taken as the problems felt by the
caretakers of the PWD. We just describe the FRM model and proceed on to justify why FRM
model is used in this study.
1ReceivedDecember8,2008. Accepted January6,2009.
2 VasanthaKandasamy,PraveenPrakashandThirusangu
§2. Description of FRM Model and its Application to the Problem
Fuzzy Relational Maps (FRMs) are constructed analogous to FCMs. FRMs are divided into
two disjoint units. We denote by R the set of nodes R ,··· ,R of the range space where
1 m
R = {(x ,··· ,x )|x = 0 or 1} for j = 1,2,··· ,m. D ,··· ,D denote the nodes of the
j 1 m j 1 n
domain space where D = {(y ,··· ,y )|y = 0 or 1} for i = 1,2,··· ,n. Here, y = 0 denotes
i 1 n i i
the off state and y =1 the on state of any state vector. Similarly x =1 denotes the on state
i i
and x =0 the off state of any state vector.
i
ThusaFRMisadirectedgraphoramapfromD toRwithconceptslikepoliciesorevents
etc as nodes and causalities as edges. It represents causal relations between the spaces D and
R.
Let D andR denote the nodes of anFRM. The directed edge fromD to R denotes the
i j i j
causality of D on R called relations. Every edge in the FRM is weighted with a number of
i j
the set {0,+1}. Let e be the weight of the edge D R ; e ∈{0,+1}. The weight of the edge
ij i j ij
D R is positive if increase in D implies increase in R or decrease in D implies decrease in
i j i j i
R i.e., causality of D on R is 1. If e = 0 then D does not have any effect on R . When
j i j ij i j
increaseinD impliesdecreaseinR ordecreaseinD impliesincreaseinR thenthecausality
i j i j
of D on R is −1.
i j
AFRMis adirectedgraphora mapfromD toR withconceptslike policiesoreventsetc,
as nodes and causalities as edges. It represents causal relations between spaces D and R.
For the FRM with D ,··· ,D as nodes of the domain space D and R ,··· ,R as the
1 n 1 n
nodes of the rangespace R,E defined as E =(e ), where e is the weightof the directed edge
ij ij
D R (or R D ); E is calledthe relationalmatrixof the FRM. A=(a ,··· ,a ), a ∈{0,1};A
i j j i 1 n i
is called the instantaneous state vector of the domain space and it denotes the on-off position
of the nodes at any instant. Similarly for the range space a = 0 if a is off and a = 1 if a is
i i i i
on. Let the edges form a directed cycle. A FRM with directed cycle is said to be a FRM with
feed back. A FRM with feed back is said to be the dynamical system and the equilibrium of
the dynamical system is called the hidden pattern; it can be a fixed point or a limit cycle.
For example let us startthe dynamicalsystemby switching onR (or D ). Letus assume
1 1
thattheFRMsettlesdownwithR andR or(D andD )oni.e.,(10000···1)or(100···01).
1 m 1 n
Then this state vector is a fixed point. If the FRM settles down with a state vector repeating
inthe form,i.e., A →A →···A →A orB →B →···→B →B ,thenthis equilibrium
1 2 i 1 1 2 i 1
is called a limit cycle.
Now we would be using FRM models to study the problem.
2.1 Justification for Using FRM
(1) We see the problems of Persons With Disability (PWD) is distinctly different from the
problems of the caretakers of the PWD. Thus at the outset we are justified in using FRM i.e.,
a set of domain attributes and a set of range attributes.
(2)Alltheattributesunderstudycannotbequantifiedasnumbers. Sothedataisoneinvolving
alargequantityoffeelings. Hencefuzzymodelsisthebestsuited,asthedataisanunsupervised
one.
StudyoftheProblemsofPersonswithDisability(PWD)UsingFRMs 3
(3) Also this model alone can give the effect of problems faced by the caretakerson the PWDs
and vice versa. So this model is best suited for our problem.
(4) Finally this model gives hidden pattern i.e., it gives a pair of resultant state vectors i.e.,
hidden pattern related with the PWDs as well as hidden pattern related with the caretakers.
Thus we use this model to analyze the problem.
Now the attributes related with the PWDs are taken as the domain space of the FRM
andthe attributes relatedwith the caretakersof the PWDs are takenas the rangespaceof the
FRM.WeshalldescribeeachoftheattributesrelatedwiththePWDsandthatofthecaretakers
in a line or two.
2.2 Attributes Related with the PWDs
Thefollowingattributesaregivenbyanexpert. TheproblemsofPWDsaretakenasthenodes
of the domain space and the attributes associated with the close caretakers are taken as the
nodes of the range space. The attributes associatedwith the PWDs are givenbelow. They are
in certain cases described in line or two.
D – Depressed. From the survey majority of the PWDs looked and said they were de-
1
pressed because of their disability and general treatment.
D – Suffer from inferiority complex.
2
D – Mental stress/agony - They often were isolated and sometimes kept in a small hut
3
outside the housewhichmade them feelsadaswellas gavetime to think about their disability
with no other work. So they were often under stress and mental tension.
D – Self Image - Majority did not possess any self image. It was revealed from the
4
discussions and survey.
D – Happy and contended.
5
D – Uninterested in life.
6
D – Dependent on others for every thing.
7
D – Lack of mobility.
8
D – Illtreated by close relatives.
9
NowtheattributesD ,D ,··· ,D aretakenasthenodesofthedomainspaceoftheFRM.
1 2 9
We give the attributes associated with the range space.
R – Poor. So cannot find money to spend on basic requirements. The PWDs go to work
1
for their livelihood.
R – Ashamed - relatives were ashamed of the PWDs.
2
R – Indifferent - They were treated indifferently by their caretakers.
3
R – PWDs are a burden to them. So they neglected them totally.
4
R – Fatalism - They said it was fate that they have a PWD as their child / relative.
5
R – Sympathetic.
6
R – Caring.
7
R – Show hatred towards the PWDs.
8
R – The caretakers were not interested in marrying them off.
9
R – The PWDs are an economic burden to them.
10
4 VasanthaKandasamy,PraveenPrakashandThirusangu
R – They were isolated from others for reasons best known to the caretakers.
11
Thus R ,R ,··· ,R are taken as the nodes of the range space of the FRM.
1 2 11
The directed graph related with the FRM is shown in Fig.2.1, in which we have omitted
the direction D →R on each edge D R for simplicity.
i j i j
D D D D D D D D D
1 2 3 4 5 6 7 8 9
R R R R R R R R R R R
1 2 3 4 5 6 7 8 9 10 11
Fig.2.1
Let the relation matrix associated with the directed graph be given by T, where T is a
9×11 matrix with entries from the set {0,−1,1}following.
0 0 1 1 0 0 0 0 1 0 0
 0 0 0 1 0 0 −1 0 0 0 1 
 0 0 1 0 0 −1 −1 1 1 0 1 
 
 
 −1 0 −1 −1 0 1 1 −1 0 −1 1 
 
 
T = −1 −1 −1 −1 0 1 1 −1 0 0 −1 
 
 
 0 1 1 1 1 −1 0 0 0 0 0 
 
 
 0 0 0 0 0 0 0 1 0 1 0 
 
 
 0 0 0 1 0 0 0 0 1 0 1 
 
 
 0 1 0 0 0 0 0 0 0 1 0 
 
 
Now we study the effect of the state vectors on the dynamical system T.
Suppose the expert wishes to study the on state of the node D and all other nodes are
1
in the off state. Let the state vector be X = (100000000). The effect of X on the dynamical
system T is given by
XT =(00110000100)=Y(say),
YTt =(312 −2 −2 2020)→(111001010)=X (say),
1
where → denotes that the resultant state vector YTt is updated and thresholded, i.e., all
negative values and 0 are replacedby 0 and all positive values greaterthan or equal to one are
StudyoftheProblemsofPersonswithDisability(PWD)UsingFRMs 5
replaced by 1. By updating we mean the co ordinate which we started in the on state should
remain in the on state till the end.
Now we find that
X T →(01111001101)=Y (say),
1 1
Y Tt →(111001111)=X (say),
1 2
X T →(01111001111)=Y (say),
2 2
Y Tt →(111001111)=X (say)=X2.
2 3
Thus the hidden pattern gives a fixed pair given by {(111001111),(01111001111)}.
Thus when the node depressed alone in the domain space is in the on state we see
this makes the nodes D ,D ,D ,D ,D ,D to come to on state in the domain space and
2 3 6 7 8 9
R ,R ,R ,R ,R , R ,R and R in the on state in the range space.
2 3 4 5 8 9 10 11
Thus wesee exceptthe nodesthe PWD hasselfimageandshe/heis happyandcontended
all other nodes come to on state. Thus this reveals if a PWD is depressed certainly he has no
self image and he is not happy and contended. Further it also reveals from the state vector in
the domain space poverty is not a cause of depression for R is in the off state. Also R and
1 6
R alone do not come to on state which clearly shows that the caretakers are not sympathetic
7
and caring which is one of the reasonsfor the PWDs to be depressed. Thus we see allnegative
attributes come to on state in both the spaces when the PWD is depressed.
Next the expertis interestedin studying the effect of the on stateof the node in the range
space viz. R i.e., the caretakers are sympathetic towards the PWDs. Let Y =(00000100000)
6
be the state vector of the range space. To study the effect of Y on the dynamical system Tt.
YTt →(000110000)=X (say),
1
X T →(00000110000)=Y (say),
1 1
Y Tt →(000110000)=X (say).
1 2
But X =X . Thus we see the hidden pattern of the state vector is a fixed pair of points
2 1
given by {(00000110000),(000110000)}. It is clear when the PWD is treated with sympathy it
makes him feel their caretakersare caring. So R come to on state. On the other hand, we see
1
she/he is happy and contended with a self image. Next the expert wishes to find the hidden
pattern of the on state of the domain node D i.e., self image of the PWD alone is in the on
4
state.
Let P =(000100000)be the given state vector. The effect of P on T is given by
PT →(00000110000)=S (say),
1