The concept of fuzzy set was developed in 1965. Since then, researchers have used this key set in many disciplines. As a brand-new conceptual system to support human-centric framework, fuzzy set has proved quite promising and effective in modeling human involvement in human-based intelligence to attain modernity in many departments like data analyzing, data mining, image coding and explaining, as well as in intelligence systems. Fuzzy set has also become an acknowledged research subject in both pure as well as in applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications. Despite being a core subject for many years, fuzzy set still attracts researchers for putting forth solutions for prime issues with certain features questioned by these notions. Fuzzy set can effectively deal with a wide spectrum of problems of the physical world via cooperation, which may be beyond the capability of classical techniques. This means the fuzzy set could have the ability to handle a wide range of problems, for instance, decision making, intelligent data analysis, processing information, pattern recognition, and optimization.

The goal of this Special Issue is to dive deeper into the new trends of fuzzy set theory and the extension of fuzzy set theory with applications in group theory, ring theory, statistics, topological spaces, graph theory, decision making and other engineering applications.

Potential topics include but are not limited to the following:

• Fuzzy sets and their extensions
• Fuzzy subgroups and their extended forms
• Applications of fuzzy sets in ring theory
• Fuzzy sets and their extensions in graph theory
• Statistics on the application of fuzzy informatics
• Fuzzy sets and their extensions in topological spaces
• Decision making application with fuzzy logic

fuzzy logic, in mathematics, a form of logic based on the concept of a fuzzy set. Membership in fuzzy sets is expressed in degrees of truth—i.e., as a continuum of values ranging from 0 to 1. In a narrow sense, the term fuzzy logic refers to a system of approximate reasoning, but its widest meaning is usually identified with a mathematical theory of classes with unclear, or “fuzzy,” boundaries. Control systems based on fuzzy logic are used in many consumer electronic devices in order to make fine adjustments to changes . Fuzzy logic concepts and techniques have also been profitably used in linguistics, the behavioral sciences, the diagnosis of certain diseases, and even  analysis.

Fuzzy sets

Most concepts used in everyday language, such as “high temperature,” “round face,” or “aquatic animal,” are not clearly defined. In 1965 Lotfi Zadeh, an engineering professor at the University of California at Berkeley, proposed a mathematical definition of those classes that lack precisely defined criteria of membership. Zadeh called them fuzzy sets. Membership in a fuzzy set may be indicated by any number from 0 to 1, representing a range from “definitely not in the set” through “partially in the set” to “completely in the set.” For example, at age 45 a man is neither very young nor very old. This makes it difficult in traditional logic  to say whether or not he belongs to the set of “old persons.” Clearly he is “sort of” old, a qualitative assessment that can be quantified by assigning a value, or degree of membership, between 0 and 1—say 0.30—for his inclusion in a fuzzy set of old persons.

 

Catalan number

   In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894).

The nth Catalan number can be expressed directly in terms of binomial coefficients The first Catalan numbers for n = 0, 1, 2, 3, ... are

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...

Applications in combinatorics

There are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers. Following are some examples, with illustrations of the cases C₃ = 5 and C₄ = 14

• C_n is the number of Dyck words^[2] of length 2n. A Dyck word is a string consisting of n X's and n Y's such that no initial segment of the string has more Y's than X's. For example, the following are the Dyck words of length 6:           XXXYYY     XYXXYY     XYXYXY     XXYYXY     XXYXYY.

• Re-interpreting the symbol X as an open parenthesis and Y as a close parenthesis, C_n counts the number of expressions containing n pairs of parentheses which are correctly matched
•                     ((()))     ()(())     ()()()     (())()     (()())

• C_n is the number of different ways n + 1 factors can be completely parenthesized (or the number of ways of associating n applications of a binary operator, as in the matrix chain multiplication problem). For n = 3, for example, we have the following five different parenthesizations of four factors:       ((ab)c)d     (a(bc))d     (ab)(cd)     a((bc)d)     a(b(cd))

Also, the interior of the correctly matching closing Y for the first X of a Dyck word contains the description of the left subtree, with the exterior describing the right subtree.

• C_n is the number of non-isomorphic ordered (or plane) trees with n + 1 vertices.^[3] See encoding general trees as binary trees.
• C_n is the number of monotonic lattice paths along the edges of a grid with n × n square cells, which do not pass above the diagonal. A monotonic path is one which starts in the lower left corner, finishes in the upper right corner, and consists entirely of edges pointing rightwards or upwards. Counting such paths is equivalent to counting Dyck words: X stands for "move right" and Y stands for "move up".

The following diagrams show the case n = 4:

This can be represented by listing the Catalan elements by column height:

The dark triangle is the root node, the light triangles correspond to internal nodes of the binary trees, and the green bars are the leaves.

[0,0,0,0] [0,0,0,1] [0,0,0,2] [0,0,1,1]

[0,1,1,1] [0,0,1,2] [0,0,0,3] [0,1,1,2] [0,0,2,2] [0,0,1,3]

[0,0,2,3] [0,1,1,3] [0,1,2,2] [0,1,2,3]

• A convex polygon with n + 2 sides can be cut into triangles by connecting vertices with non-crossing line segments (a form of polygon triangulation). The number of triangles formed is n and the number of different ways that this can be achieved is C_n. The following hexagons illustrate the case n = 4:

• C_n is the number of stack-sortable permutations of {1, ..., n}. A permutation w is called stack-sortable if S(w) = (1, ..., n), where S(w) is defined recursively as follows: write w = unv where n is the largest element in w and u and v are shorter sequences, and set S(w) = S(u)S(v)n, with S being the identity for one-element sequences.
• C_n is the number of permutations of {1, ..., n} that avoid the permutation pattern 123 (or, alternatively, any of the other patterns of length 3); that is, the number of permutations with no three-term increasing subsequence. For n = 3, these permutations are 132, 213, 231, 312 and 321. For n = 4, they are 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312 and 4321.
• C_n is the number of noncrossing partitions of the set {1, ..., n}. A fortiori, C_n never exceeds the nth Bell number. C_n is also the number of noncrossing partitions of the set {1, ..., 2n} in which every block is of size 2. The conjunction of these two facts may be used in a proof by mathematical induction that all of the free cumulants of degree more than 2 of the Wigner semicircle law are zero. This law is important in free probability theory and the theory of random matrices.
• C_n is the number of ways to tile a stairstep shape of height n with n rectangles. Cutting across the anti-diagonal and looking at only the edges gives full binary trees. The following figure illustrates the case n = 4:

AN INTRODUCTION OF SUMUDU TRANSFORM

 AN INTRODUCTION OF SUMUDU

TRANSFORM

S. Jeevitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Komala

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Silambarasi

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Susitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

R.Vanitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

ABSTRACT

In this paper, we see the definition, some basic properties and fundamental properties

of Sumudu transform, relationships between Laplace and Sumudu transforms and Existence

of Sumudu transform.

KEYWORDS

Sumudu Transform, Gamma Function, Laplace Transform.

INTRODUCTION

The Sumudu transform is introduced by Watugula. Sumudu transform may be used

to solve problems without resorting to a new frequency domain .Due to its simple

formulation and consequent special and useful properties, the Sumudu transform has

already shown much promise. It is revealed here in and elsewhere that it can help to solve

intricate problems in engineering mathematics and applied sciences. However, despite the

potential presented by this new operator, only few theoretical investigations have appeared

in the literature, over a fifteen-year period. Most of the available transform theory books, if

not all, do not refer to the Sumudu transform. Even in relatively recent well known

comprehensive handbooks, such as Debnath and Poularikas, no mention of the Sumudu

transform can be found.

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SUMUDU TRANSFORM

Watugala introduced a new transform and named as Sumudu transform which is

defined by the following formula

𝐹(𝑢) = 𝒮[𝑓(𝑡); 𝑢] =

1

𝑢

∫ 𝑒−(

1

𝑢

) 𝑓(𝑡)𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2)

∞

0

BASIC SUMUDU TRANSFORM PROPERTIES

Sumudu transform for 𝑓 ∈ 𝐴:

𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

0

Duality with Laplace transforms:

𝐺(𝑢) =

𝐹(1

⁄𝑢)

𝑢

, 𝐹(𝑠) =

𝐺(1

⁄𝑠)

𝑠

Linearity Property:

𝒮[𝑎𝑓(𝑡) + 𝑏𝑔(𝑡) = 𝑎𝒮[𝑓(𝑡)] + 𝑏𝒮[𝑔(𝑡)]]

Sumudu Transform of Function Derivatives:

𝐺1(𝑢) = 𝒮[𝑓,(𝑡)] =

𝐺(𝑢)−𝑓(0)

𝑢

=

𝐺(𝑢)

𝑢

−

𝑓(0)

𝑢

𝐺2(𝑢) = 𝒮[𝑓,,(𝑡)] =

𝐺(𝑢) − 𝑓(0)

𝑢2 =

𝐺(𝑢)

𝑢2 −

𝑓(0)

𝑢2 −

𝑓,(𝑜)

𝑢

𝐺𝑛(𝑢) = 𝒮[𝑓𝑛(𝑡)] =

𝐺(𝑢)

𝑢𝑛 −

𝑓(0)

𝑢𝑛 − ⋯ −

𝑓𝑛−1(𝑜)

𝑢

Sumudu transform of integral of a function:

𝒮 [∫ 𝑓(𝜏)𝑑𝜏

1

0 ] = 𝑢𝐺(𝑢)

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SUMUDU TRANSFORM FUNDAMENTAL PROPERTIES

THE DISCRETE SUMUDU TRANSFORM

Over the set of functions

𝐴 = {(𝑓(𝑡)|∃𝑀, 𝜏1,𝜏2 > 0, |𝑓(𝑡)| < 𝑀𝑒|𝑡|/𝜏𝑗 , 𝑖𝑓 𝑡 ∈ (−1)𝑗 × [0, ∞))}, (1)

the Sumudu transform is defined by

𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

0

(2)

Among others, the Sumudu transform was shown to have units preserving properties and

hence may be used to solve problems without resorting to the frequency domain. As will

be seen below, this is one of many strength points for this new transform, especially with

respect to applications in problems with

physical dimensions. In fact, the Sumudu transform which is itself linear, preserves linear

functions, and hence in particular does not change units (see for instance Watugala or

Belgacem et al).Theoretically, this point may perhaps best be illustrated as an implication

of this more global result.

THEOREM:1

The Sumudu transform amplifies the coefficients of the power series function,

𝑓(𝑡) = Σ 𝑎𝑛𝑡𝑛 ∞

𝑛

=0 (1.1)

by sending it to the power series function,

𝐺(𝑢) = Σ 𝑛! 𝑎𝑛𝑢𝑛 ∞

𝑛

=0 (1.2)

PROOF :

Let f (t) be in A. If 𝑓(𝑡) = Σ 𝑎𝑛𝑡𝑛 ∞

𝑛

=0 in some interval I ⊂ R, then by Taylor’s

function expansion theorem,

𝑓(𝑡) = Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑡𝑛

(1.3)

Therefore, by (2), and that of the gamma function Γ , we have

𝒮[𝑓(𝑡)] = ∫ Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 (𝑢𝑡)𝑛𝑒−𝑡 ∞

0 𝑑𝑡

= Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑢𝑛 ∫ 𝑡𝑛 ∞

0 𝑒−𝑡 𝑑𝑡

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= Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑢𝑛 Γ(n+1)

= Σ 𝑓(𝑛)(0)𝑢𝑛 ∞

𝑛

=0 (1.4)

Consequently, it is perhaps worth noting that since

𝒮[(1 + 𝑡)𝑚] = 𝒮 Σ 𝐶𝑛

𝑚𝑡𝑛 𝑚𝑛

=0

= 𝒮 Σ 𝑚!

𝑛!(𝑚−𝑛)!

𝑚𝑛

=0 𝑢𝑛

𝒮[(1 + 𝑡)𝑚] = Σ

𝑚!

(𝑚−𝑛)!

𝑚 𝑛=

0

𝑢𝑛

=Σ 𝑃𝑛

𝑚𝑢𝑛 𝑚𝑛

=0 (1.5)

the Sumudu transform sends combinations, 𝐶𝑛

𝑚 into permutations, 𝑃𝑛

𝑚, and hence may

seem to incur more order into discrete systems.

Also, a requirement that 𝒮[𝑓(𝑡)] converges, in an interval containing u=0, is provided by

the following conditions when satisfied, namely, that

(𝑖)𝑓(𝑛)(0) → 0 𝑎𝑠 𝑛 → ∞,

(𝑖𝑖) lim

𝑛→∞

|

𝑓(𝑛+1)(0)

𝑓(𝑛)(0)

𝑢| < 1 (1.6)

This means that the convergence radius r of 𝒮[𝑓(𝑡)] depends on the sequence

𝑓(𝑛)(0), since

𝑟 = lim

𝑛→∞

|

𝑓(𝑛+1)(0)

𝑓(𝑛)(0)

| (1.7)

Clearly, the Sumudu transform may be used as a signal processing or a detection

tool,especially in situations where the original signal has a decreasing power tail .

However, care must be taken, especially if the power series is not highly decaying. This

next example may instructively illustrate the stated concern. For instance, consider the

function

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𝑓(𝑡) = {

𝐼𝑛 (𝑡 + 1) 𝑖𝑓 𝑡 ∈ (−1,1]

𝑜 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(1.8)

Since 𝑓(𝑡) = Σ (−1)𝑛−1 ∞

𝑛

=1

𝑡𝑛

⁄𝑛 the expect for u=0

𝒮[𝑓(𝑡)] = Σ (−1)𝑛−1(𝑛 − 1)! 𝑢𝑛 ∞

𝑛

=1 (1.9)

Diverges throught , because its convergens radius

𝑟 = lim

𝑛→∞

|

(−1)𝑛−1(𝑛−1)!

(−1)𝑛𝑛!

|

lim

𝑛→∞

1

𝑛

= 0 (1.10)

RELATION BETWEEN SUMUDU AND LAPLACE TRANSFORM

In our study, we use the following convolution notation: double convolution

between two continuous functions F (x, y) and G(x, y) given by

𝐹1(𝑥, 𝑦) ∗∗ 𝐹2(𝑥, 𝑦) = ∫ ∫ 𝐹1(𝑥 − 𝜃1, 𝑦 − 𝜃2)𝐹2(𝜃1, 𝜃2)𝑑𝜃1𝑑𝜃2

𝑥

0

𝑦

0

The single Sumudu transform is defined over the set of the functions

𝐴 = {(𝑓(𝑡)|∃𝑀, 𝜏1,𝜏2 > 0, |𝑓(𝑡)| < 𝑀𝑒|𝑡|/𝜏𝑗 , 𝑖𝑓 𝑡 ∈ (−1)𝑗 × [0, ∞))}

by 𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

0

A sufficient condition for the existence of the Sumudu transform of a

function f is of exponential order, that is, there exist real constants

M > 0, 𝐾1, and 𝐾2 , such that |𝑓(𝑡, 𝑥)| ≤ 𝑀𝑒

𝑡

𝐾1

+

𝑥

𝐾2

EXISTENCE OF THE SUMUDU TRANSFORM

THEOREM:2

If f is of exponential order, then its Sumudu transform 𝒮[𝑓(𝑡, 𝑥)] = 𝐹(𝑣, 𝑢)exists and is

given by

𝐹(𝑣, 𝑢) = ∫ ∫ 𝑒−

𝑡

𝑣

−

𝑥

𝑢𝑓(𝑡, 𝑥)

∞

0

∞

0

𝑑𝑡𝑑𝑥

© 2021 JETIR July 2021, Volume 8, Issue 7 www.jetir.org (ISSN-2349-5162)

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where

1

𝑢

=

1

ɳ

+

𝑖

𝜏

,

1

𝑣

=

1

𝜇

+

𝑖

𝜉

The defining integral for F exists at points

1

𝑢

+

1

𝑣

=

1

ɳ

+

1

𝜇

+

𝑖

𝜏

+

𝑖

𝜉

in the right half plane

1

ɳ

+

1

𝜇

>

1

𝐾1

+

1

𝐾2

.

PROOF:

Using

1

𝑢

=

1

ɳ

+

𝑖

𝜏

𝑎𝑛𝑑

1

𝑣

=

1

𝜇

+

𝑖

𝜉

We can express 𝐹(𝑣, 𝑢) as

𝐹(𝑣, 𝑢) = ∫ ∫ 𝑓(𝑡, 𝑥)𝑐𝑜𝑠(

𝑡

𝜏

+

𝑥

𝜉

)𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

0

∞

0

𝑑𝑡𝑑𝑥

−𝑖 ∫ ∫ 𝑓(𝑡, 𝑥)𝑠𝑖𝑛(

𝑡

𝜏

+

𝑥

𝜉

)𝑒

−

𝑡

ɳ

−

𝑥

𝜇

∞

0

∞

0 𝑑𝑡𝑑𝑥

Then for values of 1

ɳ

+

1

𝜇

>

1

𝐾1

+

1

𝐾2

we have

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑐𝑜𝑠(

𝑡

𝜏

+

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

0

∞

0

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

1

𝐾1

−

1

ɳ

)𝑡+(

1

𝐾2

−

1

𝜇

)𝑥

∞

0

∞

0

𝑑𝑡𝑑𝑥

≤ 𝑀 (

ɳ𝐾1

ɳ−𝐾1

) (

ɳ 𝐾2

𝜇− 𝐾2

)

and

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑠𝑖𝑛(

𝑡

𝜏

+

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

0

∞

0

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

1

𝐾1

−

1

ɳ

)𝑡+(

1

𝐾2

−

1

𝜇

)𝑥

∞

0

∞

0

𝑑𝑡𝑑𝑥

≤ 𝑀 (

ɳ𝐾1

ɳ−𝐾1

) (

ɳ 𝐾2

𝜇− 𝐾2

)

which imply that the integrals defining the real and imaginary parts of F exist for value of

𝑅𝑒 (

1

𝑢

+ 1

𝜇

) > 1

𝐾1

+

1

𝐾2

, and this completes the proof.

Thus, we note that for a function f, the sufficient conditions for the existence of the

Sumudu transform are to be piecewise continuous and of exponential order.

We also note that the double Sumudu transform of function f(t, x) is defind by

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𝐹(𝑣, 𝑢) = 𝒮2[𝑓(𝑡, 𝑥); (𝑣, 𝑢)] =

1

𝑢𝑣

∫ ∫ 𝑒−(

𝑡

𝑣

+

𝑥

𝑢

)𝑓(𝑡, 𝑥)

∞

0

∞

0

𝑑𝑡𝑑𝑥

where, 𝑠2 indicates double Sumudu transform and 𝑓(𝑡, 𝑥 ) is a function which can be

expressed as a convergent infinite series. Now, it is well known that the derivative of

convolution for two functions 𝑓 and 𝑔 is given by

𝑑

𝑑𝑥

(𝑓 ∗ 𝑔)(𝑥) =

𝑑

𝑑𝑥

𝑓(𝑥) ∗ 𝑔(𝑥)𝑜𝑟 𝑓(𝑥) ∗

𝑑

𝑑𝑥

𝑔(𝑥)

and it can be easily proved that Sumudu transform is

𝒮 [

𝑑

𝑑𝑥

(𝑓 ∗ 𝑔)(𝑥); 𝑣] = 𝑢𝒮 [

𝑑

𝑑𝑥

𝑓(𝑥); 𝑢] 𝒮[𝑔(𝑥); 𝑢] or

= 𝑢𝒮[𝑓(𝑥); 𝑢]𝒮 [

𝑑

𝑑𝑥

𝑔(𝑥); 𝑢].

The double Sumudu and double Laplace transforms have strong relationships that may be

expressed either as

(𝐼) 𝑢𝑣𝐹(𝑢, 𝑣) = £2 (𝑓(𝑥, 𝑦); (

1

𝑢

,

1

𝑣

))

Or (𝐼𝐼) 𝑝𝑠𝐹(𝑝, 𝑠) = £2 (𝑓(𝑥, 𝑦); (

1

𝑝

,

1

𝑠

))

where £2 represents the operation of double Laplace transform. In particular, the double

Sumudu and double Laplace transforms interchange the image of sin(x + t) and cos(x + t).

It turns out that

𝑠2[sin(𝑥 + 𝑡)] = £2[cos(𝑥 + 𝑡) =

𝑢 + 𝑣

(1 + 𝑢)2(1 + 𝑣)2

And

𝑠2[cos(𝑥 + 𝑡)] = £2[sin(𝑥 + 𝑡) =

1

(1 + 𝑢)2(1 + 𝑣)2

REFERENCE:

1. G. K. Watugala, Sumudu Transform: a new integral transform to solve differential

equations and control engineering problems, Internat. J. Math. Ed. Sci.Tech. 24 (1993) 35-

43.

2.G. K. Watugala, The Sumudu transform for functions of two variables, Math.

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JETIR2107602 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org e778

Engrg. Indust. 8 (2002) 293-302.

3. M.A.Asiru,Sumudu transform and the solution of integral equations of onvolution type,

International Journal of Mathematical Education in Science and Technology 32 (2001),

no. 6, 906–910.

4. Further properties of the Sumudu transform and its applications, International Journal of

Mathematical Education in Science and Technology 33 (2002), no.3, 441–449.

5. Classroom note: application of the Sumudu transform to discrete dynamic systems,

International Journal of Mathematical Education in Science and Technology 34 (2003),

no. 6,944–949.

A STUDY ON IMPACT OF ONLINE BUYING (WITH SPECIAL REFERENCE TO EDUCATED WOMEN EMPLOYEES IN VANIYAMBADI TOWN)

 A STUDY ON IMPACT OF ONLINE BUYING

(WITH SPECIAL REFERENCE TO EDUCATED

WOMEN EMPLOYEES IN VANIYAMBADI TOWN)

N.Mahalakshmi Msc.,M.Phil, Assistant Professor,PG & Research Department of

Mathematics,

K.Rajeswari, Msc.,M.Phil, Head & Assistant Professor, Department of Statistics,

G.Shenbagavalli, Msc.,M.Phil, Assistant Professor,PG & Research Department of

Mathematics,

S.Geethalakshmi, Msc.,M.Phil, Assistant Professor,PG & Research Department of

Mathematics,

K.Valarmathy, Msc.,M.Phil, Assistant Professor,PG & Research Department of

Mathematics

ABSTRACT

Online shopping allows consumers to buy directly any goods or services from the

provider to buyer. Internet using web browser is used to communicate both the buyer and

seller. Consumer can visit the website and shall select their products. There are many

shopping zones which provide various kinds of products and services to the consumer. The

consumers shall select, compare and can see reviews from other buyers and can place their

order through online .In this paper the researcher study the impact of online buying and also to

give possible solution for the problem aroused on online buying.

KEYWORDS: Online, Consumer, Website, Shopping zone.

INTRODUCTION

English entrepreneur Michal Aldrich invented on line shopping in 1979.Online

shopping is a form of electronic commerce which allows consumer to direct them to buy

goods or service from a web browser. E-web-store, e-shop, e-store, Internet shop are the

alternative names.

From the online buying the consumers shall buy the goods for the satisfaction. In this

type of service, the time saved and varieties of products can visualize. It is a paying way for

Cash transaction.

OBJECTIVES

 To study the impact of online buying with special reference to Vaniyambadi Town.

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 To provide possible solution for the problem aroused in online buying.

SCOPE

 This paper will help the consumer to avoid problem aroused in online buying.

 To educate the consumer about the shopping zone.

LIMITATIONS

 Sample selected on Vaniyambadi so it cannot be generalized with other places.

 Fifteen days was the time period for research.

RESEARCH METHODOLOGY

Research is a systematic method of finding solutions to problems. In this paper we

describe a study on impact of online buying and its results are questionnaire research. This

research includes surveys and fact finding enquiries of different kinds. The major purpose of

research is applying statistical tools to researchers.

SAMPLING DESIGN

A sample design is a finite plane for obtaining a sample from given educated women

employees.

SAMPLE SIZE

Sample size is 100

i.e.) n=100

METHODS OF DATA COLLECTION

The data’s were collected through primary and secondary sources.

Primary data: Primary data are those which are collected for the first time and they are

original in character. If an individual or an office collects the data to study a particular

problem, the data are the raw materials of the enquiry. They are primary data collected by the

investigator himself to study any particular problem.

Secondary data: Secondary data are those which are already collected by someone for some

purpose and are available for the present study. For instance, the data collected during census

operations are primary data to the department of census and the same data, if used by a

research worker for some study, are secondary data.

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Nature of research:

Descriptive research, also known as statistical research, describes data and

characteristics about the population or phenomenon being studied. Descriptive research

answers the questions who, what, where, when and how.

Questionnaire

A well defined questionnaire is used effectively to gather information.

Nature of questions asked

The questionnaire consists of dichotomous, rating and ranking questions and multiple

choice questions.

Presentation of data

The data are presented through charts and tables.

Analysis of Data

Percentage Analysis

Pie Diagram

Chi square Analysis

DATA ANALYSIS & INTERPRETATIONS:

(Table-1) Respondents on the basis of income level

Source: Primary data

INTERPRETATION

The above table shows that 0% of the respondents are 𝑅𝑠 4000 − 7000 income level, 48%

are 𝑅𝑠 7000 − 10000 income level, 40% are 𝑅𝑠 10000 − 13000 income level, 8% are

𝑅𝑠 13000 − 16000 income level and 4% are more than 𝑅𝑠 16000 income level.

Income level No. of

respondent

Percentage

Rs4000-7000 0 0%

Rs7000-10000 48 48%

Rs10000-13000 40 40%

Rs13000-16000 8 8%

More than Rs16000 4 4%

Total 100 100%

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(Table-2) Respondents on the basis of products bought through online

Source: Primary data

INTERPRETATION

The above table shows that 112.5° of the respondents are Mobile, 67.5° are Accessories,

22.5° are Books and 157.5° are of Cloths.

(Table-3) Respondents on the basis of shopping zones and products

Observed Data

S. No Product

items

Shopping Zones Total

Amazon Flip

kart

Snap

deal

1 Mobile 10 8 2 20

2 Accessories 8 8 2 18

3 Cloths 22 3 1 26

Total 40 19 5 64

Expected Data

S. No

Product

items

Shopping Zones Total

Amazon Flip kart Snap deal

1 Mobile 12.5 5.93 1.56 19.99

2 Accessories 11.25 5.34 1.40 17.99

3 Cloths 16.25 7.7 2.03 25.98

Total 40 18.97 4.99 63.96

Source: Primary data

S. No Items No. of respondent Degree

1 Mobile 20 112.5°

2 Accessories 12 67.5°

3 Books 4 22.5°

4 Cloths 28 157.5°

Total 64 360°

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INTERPRETATION

From the above table the respondents are more interested towards buying cloths.

H0: Consumers prefer all online services for their products purchase.

H1: Consumers prefer specific branded online services for their products purchase.

Statistical Analysis

In Chi - square table

Calculated value 𝜒2 = 8.80

Degrees of Freedom

ᵞ = {(𝑐 − 1)(𝑟 − 1)} − 2

= { (3 − 1)(3 − 1) } − 2

= { (2)(2) } − 2

= 4 − 2

= 2

Table value

At 5% level of significance for 2 degrees of freedom the table value of 𝜒2 𝑖𝑠 5.991.

So H0 is rejected .Because calculated value is more than table value.

We accept H1. Consumers prefer specific branded online services for their products

purchase.

(Table-4) Respondents on the basis of shopping zones provides online services

S. No Shopping Zones No. of Respondent Percentage

1 Amazon 48 75%

2 Flip kart 4 6.25%

3 Snap deal 12 18.75%

Total 64 100%

Source: Primary data

INTERPRETATION

The above table shows that 75% of the respondents are Amazon 6.25% are Flip kart and

18.75% are Snap deal.

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(Table-5) Respondents on the basis of Problems in online buying

S. No. Problems in Online

buying

No. of Respondent Percentage

1 Products are very

less in weight

8 12.5%

2 Some are broken 24 37.5%

3 They are changing

the product items

12 18.75%

4 It is not up to the

quality mentioned

12 18.75%

5 Other reasons 8 12.5%

Total 64 100%

Source: Primary data

INTERPRETATION

The above table shows that 12.5% of the respondents are Products are very less in weight,

37.5% are some are broken, 18.75% are changing the product items, 18.75% are not up to

the quality mentioned and 12.5% are Other Reasons.

(Table-6) Respondents on the basis of possible solution for online buying

S. No. Possible solution

for Online

No. of Respondent Degree

1 Direct purchasing 12 67.5°

2 Reference

Purchasing

8 45°

3 Purchase based on

reviews and reality

28 157.5°

4 Return and refund

policy

16 90°

Total 64 360°

Source: Primary data

INTERPRETATION

The above table shows that 67.5° of the respondents are Direct Purchasing, 45° are Reference

Purchasing, 157.5° are Purchase based on reviews and reality and 90° are Return and refund

policy.

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(Table-7) Respondents on the basis of online service providers

Approaching the customers

S. No. Providers

approaching the

consumers

No. of

Respondent

Percentage

1 Mail 8 12.5%

2 SMS 8 12.5%

3 Advertising through

media

40 62.5%

4 Others 8 12.5%

Total 60 100%

Source: Primary data

INTERPRETATION

The above table shows that 12.5% of the respondents are Mail, 12.5% are SMS, 62.5% are

advertising through media and 12.5% are others.

FINDING

 Majority of the working women earn 𝑅𝑠. 7000 − 10000 as their monthly income.

 Majority of the women employees buy mobile phone through Online Services.

 Consumers prefer specific branded online services for their products purchase.

 Majority of the online purchaser are carried on Amazon. In

 Women employees are also faced with many problems in online buying. They faced

majority broken products through online purchase.

 Problem shall be solved through rectify the reviews from other buyers.

 Majority of the online service providers are advertising their products through media.

CONCLUSION

Online shopping allows consumers to buy directly any goods or services from the

provider to buyer. Internet using web browser is used to communicate both the buyer and

seller. Consumer can visit the website and shall select their products. Consumers prefer online

buying for their purchase.

Even though there are lot of problems in Online buying that can be rectified through

some measures that has discussed in the Data Interpretation part of this paper. Consumers are

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much involved towards selection of online service providers. They are not placing their orders

to all the online service providers.

They select any particular specific online service provider and they are placing their

order in that specific shopping zone that is found and justified by the researcher through chi -

square analysis.

Reference:

1. C.R.Kothari - Research Methodology and Techniques - Wishwa Prakasan-2nd Edition

2003

2. P.Ravilochan - Research Methodology - Margham Publication - 2nd Edition 2011.

3. S.Pillai and Bhagavathy – Statistics-S. Chand & Company LTD.- 7th Edition 2012.

1. NAME:

2. QUALIFICATION:

3. NATURE OF WOMEN : WORKING/ NON WORKING

4. YEAR OF EXPERIENCE:

A) 1-2 Years B) 2-3 Years C) 3-4 Years D) 4-5 Years E) more than 5 Years

5. Income level

A) Rs.4000-7000 B) Rs.7000-10000 C) Rs.10000-13000 D) Rs.13000-16000

E) more than Rs.16000

6. Are you online buyer?

A) YES B) NO

IMPACT OF PUBLIC LIBRARY AND ITS USAGE TOWORDS GENERAL PEOPLE

 IMPACT OF PUBLIC LIBRARY AND ITS  USAGE TOWORDS GENERAL PEOPLE

(WITH SPECIAL REFERENCE TO VANIYAMBADI TOWN)

K.Rajeswari.,Msc.,M.Phil, Head & Assistant professor Department of Statistics,

N.Mahalakshmi Msc.,M.Phil, Assistant Professor,PG & Research Department of

Mathematics,

K.Sulochana Msc.,M.Phil, Assistant Professor, Department of Statistics,

S. Revathi Msc.,M.Phil, Assistant Professor, Department of Statistics,

M.Sathiyapriya, Msc.,M.Phil, Assistant Professor,PG & Research Department of

Mathematics

Library

The word "library" seems to be used in so many different aspects now, from the brick-and-mortar

public library to the digital library. Public libraries and indeed, all libraries are changing and dynamic

places where librarians help people find the best source of information whether it's a book, a web site, or

database entry. The inscribed clay tablets were used nearly 5,000 years ago, as early as 3020 B.C.

ABSRACT

This paper tries to bring out the impact of public library and its usage towards general people.

Kew words: Library ,Public, Knowledge.

INTRODUCTION

Public libraries continue to be places for education and self-help and offer opportunity for people

of all ages and backgrounds. They offer opportunity for everyone to learn and to pursue selfimprovement.

In response to community needs for information, many libraries offer such programs as

English as a Second Language (ESL) classes, homework help, after-school programs for children, job

information centers, assistance for new immigrants, literacy programs, and much, much more. To serve

such community needs, public libraries collect and make available information in many, many formats.

Libraries are places where people connect not just with books and computers but with other people.

In our changing information age, we need libraries more than ever to help us sort through the information

clutter. After all, librarians are the ultimate search engine. They know how to find the best information in

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whatever form and teach others how to find, use, and evaluate information. They, of course, apply this

skill to the traditional functions of materials selection and readers advisory. The library is one of the most

valuable institutions we have, and we should not take it for granted.

 Public Library Association

A division of the American Library Association (ALA), the oldest and largest library association

in the world. Founded in 1944, PLA is a member-driven organization that exists to provide a

diverse program of communication, publication, advocacy, continuing education, and

programming for its members and others interested in the advancement of public library service.

OBJECTIVES OF THE STUDY

To find out the impact of public library and its usage towards general people.

SCOPE OF THE STUDY

This paper helps the general public to know about impact of its and its usage.

LIMITATIONS

 This study done only in Vaniyambadi town.

 Time duration of the study in only 15 days.

RESEARCH METHODOLOGY

 Research Design

Research design indicates the methods and procedures of conducting research study.

 Sampling size

80

 Data Collection Method

For primary data ,questionnaires method was followed .The questionnaires were issued to

collect the data. with Statistical tools and its results done in a formal way. So as to gather

information about the feeling of the respondent with regard to the topic under investing.

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Sources of Data

 Primary data

The primary data has been collected by means of questionnaires.

 Secondary Data

The researcher collected the secondary form research books and also gathered information by

browsing in the net.

 Percentage Analysis

Percentage bar diagram is used to highlight the relative importance of the various components

parts to the whole. The total for each bar taken as 100 and the value of each component is

expressed as percentage of the respective totals.

 Pie diagram

The pie diagram ranks high in understanding .Just as we divide a bar or a rectangle to show its

components, a circle can also be divide into sectors .As there are 360 degree at the center,

proportionate sectors are cut taking the whole data equal to 360 degrees.

 Chi-square test

Suppose we are given a set of observed frequencies obtained under some experiment and we want

to test if the experimental results support a particular hypothesis or theory. Karl Pearson in 1990,

developed a test for testing the significance of the discrepancy between experimental values and

the theoretical values obtained under some hypothesis

DATA ANALYSIS & INTERPRATION

(Table-1) Respondents of visitors in public library

Years

Respondent Percentage Cumulative

percentage

1-2 10 12.5% 12.5%

2-3 12 15% 27.5%

3-4 12 15% 42.5%

4-5 14 17.5% 60%

More than 5 32 40% 100%

Total 80 100%

Source :Primary data

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INTERPRETATION

The above table shows that 12.5% of the respondent are to visit 1-2 years,15% are 2-3 years,15% are 3-4

tears 17.5% are 4-5 years and 40% are more than 5 years.

(Table-2) Respondents for facility criteria in public library

Library facilities Respondent Degree Cumulative

degree

Reading Hall 10 450 450

Different Books 15 67.50 1120.5

Newspaper &

Magazine

10 450 157.50

Others 5 22.50 1800

All of the above 40 1800 3600

Total 80 3600

Source :Primary data

INTERPRETATION

The above table shows that facility criteria in public library 450 of the respondent are reading hall, 67.50

are different books , 450 are Newspaper & Magazine 22.50 facility and 1800 are all of the above facilities.

(Table-3) Available of Books and their opinions

Observed data

BOOKS AVAILABLE OPENIONS TOTAL

EXCELLAENT SATISFACTORY POOR

HISTORICAL NOVEL 8 5 2 15

STORY BOOKS 11 9 3 23

BIOGRAPHY 8 7 2 17

SOCIAL & ETHICS

BOOKS

9 4 4 17

OTHERS 3 3 2 8

TOTAL 39 28 13 80

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Expected data

In χ2 calculation table

χ2 =2.632

Degrees of freedom

ɤ= {(c-1)(r-1)} -6

={ (3-1)(5-1) }-6

= { (2)(4) }-6

= {8}-6

=2

Table value

At 5% level of significance for 2 DOF the value from χ2 table 5.991

I conclude here H0 is accepted. Because calculated value not exceed than the table value.

So,There is satisfaction of studying any kind of books chosen by the readers.

FINDINGS

 Majority of the respondent are visits library for more than 5 years

 Majority of the respondent says that they are availing all the facilities in public library

 Majority of the respondent knows many aspects of public library

BOOKS AVAILABLE OPENIONS TOTAL

EXCELLAENT SATISFACTORY POOR

HISTORICAL NOVEL 7.3 5.25 2.4 14.95

STORY BOOKS 11.2 8.1 3.7 23

BIOGRAPHY 8.2 5.95 2.7 16.85

SOCIAL & ETHICS

BOOKS

8.2 5.95 2.76 16.91

OTHERS 3.9 2.8 1.3 8

TOTAL 38.8 28.5 12.86 79.71

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 Majority of the school students use to study biography and women study story books &

Historical novels

 Majority of the men are studying news paper.

 All the respondents are studying different types of books and they are well satisfied of the

books chosen by themselves and it is proved by the researcher thought chi- square

analysis.

SUGGESTION

Various kinds of books and facilities are available in library but they are not utilized by general

Public to its optimum level.

Public shall visit library and shall wait avail all of its facilities.

CONCLUSION

Various kinds of facilities are available in library and it is provided fully for general public readers

are choosing different types of books like social books, ethical books, biography, historical books, News

papers & magazines etc., The readers are choosing the books in their own interest and they are getting

satisfaction of the chosen books by them and that was proved by the researcher through chi-square

analysis .Hence on the conclusion of the paper in the researcher conclude that public library carrier

positive impact on general public.

REFERENCE

 Fundamentals of Applied Statistics -S.C.Gupta, Himalaya publishing house

 Fundamentals of mathematical Statistics -S.C.Gupta and V.K.Kapoor-sultan & sons

 Hogg, R.V. & Craig. A. T. (1998): Introduction to Mathematical Statistics, Macmillan

 Statistics-Theory & Practice, R.S.N.Pillai & Bagavathi,S.Chand

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1.NAME

2.AGE

3.QUALIFICATION

4.SEX

5.How long you go to public library

a) 1-2 b) 2-3 c) 3-4 d) 4-5 e) more than 5 years

6. What are the facilities available in public library

a)Reading hall b) Different Books c) News paper and Magazine d) Others e) All of above

7.What kind of books you read

a) Historical Novels b) Story Books c) Biography d) social and ethic books e)others

8. What are types you know

a)Mobile library b) Public library c) Private library d)Personal library e) others

9. Are you have habbit of reading books in library

a) YES B) NO

10.State the satisfaction you set out of library books.

BOOKS AVAILABLE

EXCELLAENT SATISFACTORY POOR

HISTORICAL NOVEL

STORY BOOKS

BIOGRAPHY

SOCIAL & ETHICS

BOOKS

OTHERS

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11.What kind of improvement you expect in public library

a) Arrangements of books in proper manner b)Increase no. of story books

c) can provide additional books to regular visitors d)Can Increase date return

for the books taken

e) Can provide infrastructure facility to attract students also

12.In your opinion which age group people are mostly facilitated by public library

a) 10-20 b) 20-30 c) 30-40 d) 40-50 e) Above 50

The Fibonacci Sequence:

 

The Fibonacci Sequence:

When Maths Turns Golden UNBOUND Learn how to see, and realize that everything connects to everything else: Leonardo Da Vinci Fibonacci Sequence has captivated Mathematicians, artists, designers, and scientists for centuries. Wondering what’s so special about it? Let us begin with the history. The original problem that Leonardo Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances. Suppose a newly-born pair of rabbits, one male, and one female are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month, a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one year? Think! No? Let me help you. At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month, the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. Can you see the pattern here? 1, 1, 2, 3, 5, 8, 13, 21, 34…… The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Fibonacci Sequence is a set of numbers that start with a one, followed by a one, and proceeds based on the rule that each number is equal to the sum of the preceding two numbers. The Fibonacci numbers can be thought of as Nature’s numbering system. They appear everywhere in Nature, from the leaf arrangement in plants to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. In the seeming randomness of the natural world, we can find many instances of a Mathematical order involving the Fibonacci numbers themselves and the closely related “Golden” elements. Let’s add one more interesting thing here: If we take the ratio of two successive numbers in Fibonacci’s series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers: 1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666..., 8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538... The ratio seems to be settling down to a particular value, which we call the ‘golden ratio’ or ‘the golden number’. It has a value of approximately 1·618034 and we denote it by “Phi”. Now, let’s get acquainted with some of the endless examples that make Fibonacci a wonder or ‘Golden’ sequence. Flower petals: The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five, the chicory’s 21, the daisy’s 34, and so on. Each petal is placed at 0.618034 per turn (out of a 360° circle) allowing for the best possible exposure to sunlight and other factors. Seed heads: The head of a flower is also subject to Fibonaccian processes. Typically, seeds are produced at the centre and then migrate towards the outside to fill all the space. Sunflowers provide a great example of these spiraling patterns. l know it's possible." ~ Gloria Steinem 

WOMEN OF MATHEMATICS

 

WOMEN OF MATHEMATICS

UNBOUND - archal society where the world was dictated by the likes of men, women were oppressed if they had an opinion. Obviously, a woman establishing a theorem was unheard of. However, there were a few women who dared to go against the flow and their achievements demonstrate that women have as much to contribute to Mathematics as any of their male counterparts. It is hard to perceive who the first female Mathematician was. Hypatia was certainly one of the earliest. She was born in 370 AD. She was the daughter of Theon, the last known member of the famed library of Alexandria. She followed his footsteps in the study of Mathematics and Astronomy. She collaborated with her father on commentaries of classical Mathematical works, translating them and incorporating explanatory notes, as well as creating commentaries of her own and teaching a succession of students from her home. A philosopher, a follower of Neoplatonism, a belief system in which everything emanates from the One, Hypatia was highly popular among crowds who listened to her public lectures about Plato and Aristotle. Born in an Era of revolt and revolution, Sophie Germain was born in the year 1776. Paris was exploding with the revolution when young Sophie retreated to her father’s study and began reading. After learning about Archimedes’ death, she began a lifelong study of Mathematics and Geometry, even teaching herself Latin and Greek so that she could read classic works. Unable to study at the École Polytechnique because she was female, Germain obtained lecture notes and submitted papers to Joseph Lagrange, a faculty member, under a false name. When he learned she was a woman, he became a mentor and Germain soon began corresponding with other prominent Mathematicians at the time. She became the first woman to win a prize from the French Academy of Sciences, for work on a theory of elasticity despite not having formal training and access to resources that male Mathematicians had at that time. Her proof of Fermat’s Last Theorem, though unsuccessful, was used as a foundation for work on the subject well into the twentieth century. Augusta Ada Byron, born on December 10, 1815, (later Countess of Lovelace) was brought up single-handedly by her mother after her father, poet Lord Byron was forced to leave England due to a scandal shortly after her birth. Her overprotective mother, who wanted her to grow up to be unemotional and unlike her father, encouraged her study of Science and Mathematics. As an adult, Lovelace began to correspond with the inventor and Mathematician Charles Babbage, who asked her to translate an Italian Mathematician’s memoir analyzing his Analytical Engine (a machine that would perform simple Mathematical calculations and be programmed with punch cards and is considered one of the first computers). Lovelace went beyond completing a simple translation, however, and wrote her own set of notes about the machine and even included a method for calculating a sequence of Bernoulli numbers; this is now acknowledged as the world’s first computer program. Because Russian women could not attend university, Sofia Vasilyevna (1850-1891) contracted a marriage with a young paleontologist, Vladimir Kovalevsky, and they moved to Germany. There she could not attend university lectures due to societal norms, but she was tutored privately and eventually received a doctorate after writing treatises on partial differential equations, Abelian integrals, and Saturn’s rings. Following her husband’s demise, Kovalevskaya served as a lecturer in Mathematics at the University of Stockholm and later became the first woman in that region of Europe to receive a full professorship. ‘She continued to make great strides in Mathematics, winning the Prix Bordin from the French Academy of Sciences in 1888 for an essay on the rotation of a solid body as well as a prize from the Swedish Academy of Sciences the next year’. 18 4553 By: Esha Awasthi Mathematics (Hons.) H2A In 1935, Albert Einstein wrote a letter to the New York Times, praising profusely the recently deceased Emmy Noether as “the most significant creative Mathematical genius thus far produced since the higher education of women began.” Noether had overcome many hurdles before she could collaborate with the famed physicist. She was brought up in Germany and her Mathematics education suffered a great deal because of rules against women matriculating at universities. ‘After she finally received her Ph.D., for a dissertation on a branch of abstract algebra, she was unable to obtain a university position for many years, eventually receiving the title of “unofficial associate professor” at the University of Göttingen, only to lose that in 1933 because she was Jewish.’ And so she moved to America and became a lecturer and researcher at Bryn Mawr College and the Institute for Advanced Study in Princeton, New Jersey. There she developed many of the Mathematical foundations for Einstein’s general theory of relativity and made significant advances in the field of algebra. Despite being bound by the backward, orthodox societal norms, these women showed remarkable progress in the field of Mathematics and Science. It is noteworthy how they achieved excellence in their respective fields despite lack of resources and a prevalent chauvinistic society. It is undoubtedly true that we would have never achieved the milestones of success in the field of Mathematical Science if it hadn’t been for these and hundreds of other strong-headed women who decided to defy the society for good and push the world of Science into a more progressive stage. 19 "The best way for us to cultivate fearlessness in our daughters and other young women is by example. If they see their mothers and other women in their lives going forward despite fear