# 1.1. are employed, the interval of confidence and the

1.1.
Fuzzy
Mathematical Modelling

Vagueness is embroiled in
several real phenomena. Whether one contemplates vagueness explicitly when
modeling such a phenomenon is one of the modeling choices, the outcome of which
will rest on on the situation. On the other hand, the modeler chooses to reflect
vagueness, he or she will have to select the method for modeling it. Some
experts assert that one theory, e.g. probability theory, is adequate to model
every kind of vagueness. The excellence of the techniques used in a statistical
analysis rest on extremely on the presumed probability model or distribution.
Some physical systems,   those complex
ones, are uncompromising to model by an accurate and precise mathematical
procedure or equation due to the complexity of the system structure,
nonlinearity, uncertainty, randomness, etc.

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Therefore, approximation
modeling is often necessary for real-world applications.  But, the important questions are what kind of
approximation is upright, where the logic of “goodness” has to be first
defined, of course, and how to formulate such a good approximation in modeling
a system such that it is mathematically rigorous and can produce satisfactory
results in both theory and applications.

It is clear that interval
mathematics and fuzzy logic together can provide a promising alternative to
mathematical modeling for many physical systems that are too vague or too
complicated to be described by simple and crisp mathematical formulas or
equations. When interval mathematics and fuzzy logic are employed, the interval
of confidence and the fuzzy membership functions are used as approximation
measures, leading to the so-called fuzzy systems modeling.

Zimmermann H.-J. 134
proposed a definition of uncertainty: Uncertainty implies that in a certain
situation a person does not dispose about information which quantitatively and
qualitatively is appropriate to describe, prescribe or predict
deterministically and numerically a system, its behavior or other
characteristics. Buckley 12 define fuzzy probabilities, which will be fuzzy
numbers, from a set of confidence intervals and use fuzzy numbers for the
parameters in the probability density functions, to produce fuzzy probability
density functions.

Once dealing with the usual
probability concept, an occurrence has its specific limit. For take a
situation, if an event is X= {2, 4, 7, 9, 11}, its margin is sharp and
consequently, it can be characterized as a crisp set. As soon as we deal an
occurrence whose limit is not sharp, it can be measured as a fuzzy set, that
is, a fuzzy event. We can recognize the probability of two behaviors. One is
dealing with the probability of a crisp value (crisp probability) and the other
as a fuzzy set (fuzzy probability).

A hormone activity is a very
complex system, where it secretion processing is narrowly connected to
life-sustaining developments. It is evident now that there are a number of
amino acid combinations in the hormone that keep up its activity. Some details
of hormone activity, like stress or obesity, anxiety are well understood, while
others, like fetal growth, uterus contraction during cesarean and normal
delivery, primary postpartum hemorrhage effects and many others are still
ambiguous.

This means that a reasonable
explanation of the performance of a complex organism similar to a hormone’s
function is expressed in a natural-linguistic method with “sensitive” notations
like great excitability, weak damage, the low anticipation of punishment and so
on, which cannot be through statistically assigned.So, if we need to improve an
adequate tool for the logical theory of a hormone’s effective, in some sense,
we are bound to search for a mathematical tool, which could rightly work with
“perceptions” as with mathematical objects. Such mathematical tool was proposed
by Zadeh 131 and has been further developed during the last decades. It is
called fuzzy logic and fuzzy set theory. Guanrong Chen and Trung Tat Pham 50
give an example for fuzzy rule-based health monitoring expert systems, fuzzy
logic rules to control a focus ring of a camera to allow automatic focusing for
a sharp image, application of fuzzy control to a general class of servo
mechanic systems, the fuzzy controller for the robotic manipulator. The fuzzy
terms and definitions are defined in the following section 67, 135.

1.1.1.
Fuzzy set

If X is a cluster of items indicated by x, then a fuzzy set

in X is a set of ordered pairs:

is called the membership function or grade of
membership (also degree of compatibility or degree of truth) of x in

that maps X to the membership space M=0,1. ( The membership
function is not constrained to values between 0 and 1.)

1.1.2.

Cut

The
(crisp) set of elements that belongs the the fuzzy set

at least to the degree

is called the

cut:

is called strong

cut or “strong level set”

1.1.3.
Triangular
Fuzzy Number

Among the various shapes of fuzzy number, triangular fuzzy number
is the most popular one. It is a fuzzy number represented with three points as
follows:

The
above description interpreted as membership functions in Fig. 1.1.

Fig.1.1.
Triangular fuzzy number

1.1.4.
Arithmetic
operations on fuzzy triangular number

Let A and
B be two triangular fuzzy numbers defined as

.

Some
important properties of operation of fuzzy numbers are summarized

: Triangular fuzzy number

(ii)  Subtraction

: Triangular
fuzzy number

(iii)
Symmetric image

:
Triangular fuzzy number

(iv)  The results from multiplication or division
are not triangular fuzzy numbers.

(v)   Max or min
operation does not give triangular fuzzy numbers.

1.2.Oxytocin

Oxytocin is a  mammalian neurohypophysial nonapeptide hormone
secreted by the posterior pituitary gland. Oxytocin is a nine amino acid long
peptide. The amino acid organization of oxytocin is:
cysteine-tyrosine-isoleucine-glutamine-asparagine-cysteine-proline-leucine-glycine-amide(Cys-Tyr-Ile-Gln-Asn-Cys-Pro-Leu-Gly)
and its molecular formula is C43H66N12O12S2. The molecular mass of oxytocin is
1007.187 Da.Oxytocin plays vital parts in several regulatory tasks. For
example, oxytocin performs as a neuromodulator and has been revealed to be
involved in strain, agitation, faith, benevolence, societal credit, orgasm, parturition,
lactation, maternal behaviors, mother-child, duo closeness and these protagonists
altered by variations in both oxytocin and oxytocin receptor concentrations.

The molecule structure andchemical stucture of oxytocin was presented in
Fig.1.1. and Fig. 1.2.

Fig. 1.2 Molecule
structure of oxytocin

Fig. 1.3. Chemical structure of oxytocin.

1.3.
Oxytocin
effects on social behaviour

Amid its numerous protagonists in body and brain, oxytocin
impacts societal manners. Compassionate the particular nature of this influence
is crucial, mutually within the abundant hypothetical background of
neurobiology, social neuroscience, and intellect evolution, but in the same way
within a clinical background of syndromes such as nervousness, schizophrenia,
and autism. Exploration discovering oxytocin’s role in human social behavior is
challenging due to its release in both body and brain and its interconnecting
effects with other hormones and neuromodulators.

1.4.
Oxytocin
role in love

Burman and Margolin 14
conversed the loving relations can have an intense consequence on adults’
healthiness. The central role of oxytocin in pair bonding and human imaging
studies implicate oxytocin-rich brain areas in early romantic love explained by
Inna Schneiderman et al. 58. Also, the areas of the brain are those rich in
oxytocin receptors supportive the creation of quixotic affection queried by
Acevedo et al. 1. Oxytocin stimulates a common sense of well-being comprising
calm, better social interactions, amplified faith, and condensed fear as well
as endocrine and physiological changes according to IsHak et al.126.

1.5.
The role of oxytocin in women

The release of oxytocin by
the pituitary gland acts to regulate some female reproductive functions.

1.
Stretch of the uterus and the
uterine cervix or stimulation of the breasts nipples increases action
potentials in axons of oxytocin-secreting neurons.

2.
Action potentials are conducted by
sensory neurons from the uterus and breast to the spinal cord and up ascending
tracts to the hypothalamus.

3.
Action potentials are conducted by
axons of oxytocin-secreting neurons in the hypothalamohypophysical tract to the
posterior pituitary, where they increase oxytocin secretion.

4.
Oxytocin enters the circulation,
increasing contractions of the uterus and milk ejection from the lactating
breast.

Fig. 1.3. Oxytocin role in women

Oxytocin
is a signaling substance in the brain that when released during birth, skin to skin
interaction and breast feeding induces important physiological and
psychological acclimatisation in the mother and infant. The way we give birth, feed,
and interrelate with our progenies may influence the release of oxytocin and
the development of both stop gap and enduring oxytocin-linked effects in both
mother and infant. Medicinal intrusions in birth may influence the discharge of
oxytocin and the development of the oxytocin-linked effects.

1.6.
Oxytocin
role in enhancing well-being

Oxytocin convinces a general wisdom of well-being as well as composed,
enhanced societal contact, amplified faith, and condensed distress, endocrine
and physiological changes. Some significant effects of oxytocin are momentary
and its discharge is concomitant with induction of ancillary biochemical happenings
which intervene long-term benefits together with blood pressure decline, serene
and affiliative behavior. As oxytocin release is amplified by touch and
physiological support so the hormone is involved in both source and benefits of
social interactions. Just as oxytocin has pervasive effects in factors
encompassing well-being, its dysfunction is associated with morbidity and
decreased the quality of life as observed neuropsychiatric conditions such as
autism, schizophrenia and social phobias (IsHak et al.126).