Mathematics Department: 2021

Thursday 16 December 2021

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] =

𝑢

∫ 𝑒−(

𝑢

) 𝑓(𝑡)𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2)

∞

BASIC SUMUDU TRANSFORM PROPERTIES

Sumudu transform for 𝑓 ∈ 𝐴:

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

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

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) ∞

(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

= 𝒮 Σ 𝑚!

𝑛!(𝑚−𝑛)!

𝑚𝑛

=0 𝑢𝑛

𝒮[(1 + 𝑡)𝑚] = Σ

𝑚!

(𝑚−𝑛)!

𝑚 𝑛=

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

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→∞

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

𝑦

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) ∞

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)

∞

𝑑𝑡𝑑𝑥

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where

𝑢

𝑖

𝜏

𝑣

𝜇

𝑖

𝜉

The defining integral for F exists at points

𝑢

𝑣

𝜇

𝑖

𝜏

𝑖

𝜉

in the right half plane

𝜇

𝐾1

𝐾2

PROOF:

Using

𝑢

𝑖

𝜏

𝑎𝑛𝑑

𝑣

𝜇

𝑖

𝜉

We can express 𝐹(𝑣, 𝑢) as

𝐹(𝑣, 𝑢) = ∫ ∫ 𝑓(𝑡, 𝑥)𝑐𝑜𝑠(

𝑡

𝜏

𝑥

𝜉

)𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

𝑑𝑡𝑑𝑥

−𝑖 ∫ ∫ 𝑓(𝑡, 𝑥)𝑠𝑖𝑛(

𝑡

𝜏

𝑥

𝜉

)𝑒

−

𝑡

−

𝑥

𝜇

∞

0 𝑑𝑡𝑑𝑥

Then for values of 1

𝜇

𝐾1

𝐾2

we have

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑐𝑜𝑠(

𝑡

𝜏

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

𝐾1

−

)𝑡+(

𝐾2

−

𝜇

)𝑥

∞

𝑑𝑡𝑑𝑥

≤ 𝑀 (

ɳ𝐾1

ɳ−𝐾1

) (

ɳ 𝐾2

𝜇− 𝐾2

)

and

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑠𝑖𝑛(

𝑡

𝜏

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

𝐾1

−

)𝑡+(

𝐾2

−

𝜇

)𝑥

∞

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 (

𝑢

+ 1

𝜇

) > 1

𝐾1

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)] =

𝑢𝑣

∫ ∫ 𝑒−(

𝑡

𝑣

𝑥

𝑢

)𝑓(𝑡, 𝑥)

∞

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); (

𝑢

𝑣

))

Or (𝐼𝐼) 𝑝𝑠𝐹(𝑝, 𝑠) = £2 (𝑓(𝑥, 𝑦); (

𝑝

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 + 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.

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.

Wednesday 17 November 2021

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

Thursday 21 October 2021

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

 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

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

Thursday 16 September 2021

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

Wednesday 25 August 2021

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

Wednesday 28 July 2021

Bifurcation theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.Bifurcations occur in both continuous systems (described by OD Es, DD Es or PD Es) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them with motif.

Bifurcation types

It is useful to divide bifurcations into two principal classes:

Local bifurcations, which can be analyses entirely through changes in the local stability properties of equilibrium, periodic orbits or other invariant sets as parameters cross through critical thresholds; and
Global bifurcations, which often occur when larger invariant sets of the system 'collide' with each other, or with equilibrium of the system. They cannot be detected purely by a stability analysis of the equilibrium (fixed points).
A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change. In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (those described by maps rather than OD Es), this corresponds to a fixed point having a Eloquent multiplier with modulus equal to one. In both cases, the equilibrium is non-hyperbolic at the bifurcation point. The topological changes in the phase portrait of the system can be confined to arbitrarily small neighborhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point (hence 'local').
More technically, consider the continuous dynamical system described by the ODE

${\dot {x}}=f(x,\lambda )\quad f\colon \mathbb {R} ^{n}\times \mathbb {R} \rightarrow \mathbb {R} ^{n}.$

A local bifurcation occurs at $(x_{0},\lambda _{0})$ if the Jacobin matrix ${\textrm {d}}f_{x_{0},\lambda _{0}}$ has an eigenvalue with zero real part. If the eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, but if the eigenvalue is non-zero but purely imaginary, this is a Ho pf bifurcation.
For discrete dynamical systems, consider the system

$x_{n+1}=f(x_{n},\lambda )\,.$

Then a local bifurcation occurs at $(x_{0},\lambda _{0})$ if the matrix ${\textrm {d}}f_{x_{0},\lambda _{0}}$ has an eigenvalue with modulus equal to one. If the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to −1, it is a period-doubling (or flip) bifurcation, and otherwise, it is a Hopf bifurcation.
Examples of local bifurcations include:
Saddle-node (fold) bifurcation
Transcritical bifurcation
Pitchfork bifurcation
Period-doubling (flip) bifurcation
Hopf bifurcation
Neimark–Sacker (secondary Hopf) bifurcation
Global bifurcations

A phase portrait before, at, and after a homo clinic bifurcation in 2D. The periodic orbit grows until it collides with the saddle point. At the bifurcation point the period of the periodic orbit has grown to infinity and it has become a homo clinic orbit. After the bifurcation there is no longer a periodic orbit. Left panel: For small parameter values, there is a saddle point at the origin and a limit cycle in the first quadrant. Middle panel: As the bifurcation parameter increases, the limit cycle grows until it exactly intersects the saddle point, yielding an orbit of infinite duration. Right panel: When the bifurcation parameter increases further, the limit cycle disappears completely.

Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibrium. This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighborhood, as is the case with local bifurcations. In fact, the changes in topology extend out to an arbitrarily large distance (hence 'global').
Examples of global bifurcations include:
Homo clinic bifurcation in which a limit cycle collides with a saddle point.Homo clinic bifurcations can occur super critically or sub critically. The variant above is the "small" or "type I" homo clinic bifurcation. In 2D there is also the "big" or "type II" homo clinic bifurcation in which the homo clinic orbit "traps" the other ends of the unstable and stable manifolds of the saddle. In three or more dimensions, higher co dimension bifurcations can occur, producing complicated, possibly chaotic dynamics.
Hetero-clinic bifurcation in which a limit cycle collides with two or more saddle points; they involve a hetero-clinic cycle. Hetero-clinic bifurcations are of two types: resonance bifurcations and transverse bifurcations. Both types of bifurcation will result in the change of stability of the hetero-clinic cycle. At a resonance bifurcation, the stability of the cycle changes when an algebraic condition on the eigenvalues of the equilibrium in the cycle is satisfied. This is usually accompanied by the birth or death of a periodic orbit. A transverse bifurcation of a hetero-clinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibrium in the cycle passes through zero. This will also cause a change in stability of the hetero-clinic cycle.
Infinite-period bifurcation in which a stable node and saddle point simultaneously occur on a limit cycle. As the limit of a parameter approaches a certain critical value, the speed of the oscillation slows down and the period approaches infinity. The infinite-period bifurcation occurs at this critical value. Beyond the critical value, the two fixed points emerge continuously from each other on the limit cycle to disrupt the oscillation and form two saddle points.
Blue sky catastrophe in which a limit cycle collides with a non hyperbolic cycle.
Global bifurcations can also involve more complicated sets such as chaotic at tractors

Co- dimension of a bifurcation:

The co-dimension of a bifurcation is the number of parameters which must be varied for the bifurcation to occur. This corresponds to the co-dimension of the parameter set for which the bifurcation occurs within the full space of parameters. Saddle-node bifurcations and Ho-pf bifurcations are the only generic local bifurcations which are really co-dimension-one (the others all having higher co-dimension). However, trans critical and pitchfork bifurcations are also often thought of as co-dimension-one, because the normal forms can be written with only one parameter.

An example of a well-studied co-dimension-two bifurcation is the Zhdanov–Ta-kens bifurcation.

Applications in semi classical and quantum physics:

Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems, molecular systems, and resonant tunneling diodes.Bifurcation theory has also been applied to the study of laser dynamics and a number of theoretical examples which are difficult to access experimentally such as the kicked top and coupled quantum wells. The dominant reason for the link between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits becomes large, as Martin Gutz willer points out in his classic work on quantum chaos.Many kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Ho-pf bifurcations, umbilici bifurcations, period doubling bifurcations, re-connection bifurcations, tangent bifurcations, and cusp bifurcations.

Tuesday 29 June 2021

Abstract algebra

This article is about the branch of mathematics. For the Swedish band, see Abstract Algebra.

"Modern algebra" redirects here. For van der Waerden's book, see Moderne Algebra.

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.

Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.

Early group theory

There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry.

Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory. In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work; but it appears he did not tie his definition with previous work on groups, particularly permutation groups. In 1882, considering the same question, Heinrich M. Weber realized the connection and gave a similar definition that involved the cancellation property but omitted the existence of the inverse element, which was sufficient in his context (finite group)

Modern algebra

The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an abstract group. Questions of structure and classification of various mathematical objects came to forefront.

These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. Hence such things as group theory and ring theory took their places in pure mathematics. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building up on the work of Ernst Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's Moderne Algebra, the two-volume monograph published in 1930–1931 that forever changed for the mathematical world the meaning of the word algebra from the theory of equations to the theory of algebraic structures.

Examples involving several operations include:

· Ring

· Field

· Module

· Vector space

· Algebra over a field

· Associative algebra

· Lie algebra

· Lattice

· Boolean algebra

Applications

· Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.

· In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian.

Thursday 20 May 2021

Linear Programming

BASIC CONCEPTS OF LINEAR PROGRAMMING:

In Mathematics, linear programming is a method of optimising operations with some constraints. The main objective of linear programming is to maximize or minimize the numerical value. It consists of linear functions which are subjected to the constraints in the form of linear equations or in the form of inequalities.

Linear programming is considered an important technique that is used to find the optimum resource utilisation. The term “linear programming” consists of two words as linear and programming. The word “linear” defines the relationship between multiple variables with degree one. The word “programming” defines the process of selecting the best solution from various alternatives.

Linear Programming is widely used in Mathematics and some other field such as economics, business, telecommunication, and manufacturing fields. In this article, let us discuss the definition of linear programming, its components, a simplex method with linear programming problems.

What is Linear Programming?

Linear programming (LP) or Linear Optimisation may be defined as the problem of maximizing or minimizing a linear function that is subjected to linear constraints. The constraints may be equalities or inequalities. The optimisation problems involve the calculation of profit and loss. Linear programming problems are an important class of optimisation problems, that helps to find the feasible region and optimise the solution in order to have the highest or lowest value of the function.

Linear programming is the method of considering different inequalities relevant to a situation and calculating the best value that is required to be obtained in those conditions.

Some of the assumption taken while working with linear programming are:

The number of constraints should be expressed in the quantitative terms
The relationship between the constraints and the objective function should be linear
The linear function (i.e., objective function) is to be optimised

Components of Linear Programming

The basic components of the LP are as follows:

Decision Variables
Constraints
Data
Objective Functions

Linear Programming Problems

The Linear Programming Problems (LPP) is a problem that is concerned with finding the optimal value of the given linear function. The optimal value can be either maximum value or minimum value. Here, the given linear function is considered an objective function. The objective function can contain several variables, which are subjected to the conditions and it has to satisfy the set of linear inequalities called linear constraints. In Linear programming, the term “linear” represents the mathematical relationship that is used in the given problem (Generally, linear relationship) and the term “programming” represents the method of determining the particular plan of action. The linear programming problems can be used to get the optimal solution for the following scenarios, such as manufacturing problems, diet problems, transportation problems, allocation problems and so on.

Characteristics of Linear Programming

The following are the five characteristics of the linear programming problem:

Constraints – The limitations should be expressed in the mathematical form, regarding the resource.

Objective Function – In a problem, the objective function should be specified in a quantitative way.

Linearity – The relationship between two or more variables in the function must be linear. It means that the degree of the variable is one.

Finiteness – There should be finite and infinite input and output numbers. In case, if the function has infinite factors, the optimal solution is not feasible.

Non-negativity – The variable value should be positive or zero. It should not be a negative value.

Decision Variables – The decision variable will decide the output. It gives the ultimate solution of the problem. For any problem, the first step is to identify the decision variables.