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

 

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

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

  A local bifurcation occurs at ${\displaystyle (x_{0},\lambda _{0})}$ if the Jacobin matrix ${\displaystyle {\textrm {d}}f_{x_{0},\lambda _{0}}}$ 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

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

  Then a local bifurcation occurs at ${\displaystyle (x_{0},\lambda _{0})}$ if the matrix ${\displaystyle {\textrm {d}}f_{x_{0},\lambda _{0}}}$ 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.

 

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.

 

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.

 

Cryptography

 

Cryptography, or cryptology (from Ancient Greek: κρυπτός, romanized: kryptós "hidden, secret"; and γράφειν graphein, "to write", or -λογία -logia, "study", respectively^[1]), is the practice and study of techniques for secure communication in the presence of third parties called adversaries.^[2] More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages;^[3] various aspects in information security such as data confidentiality, data integrity, authentication, and non-repudiation^[4] are central to modern cryptography. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, electrical engineering, communication science, and physics. Applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications.

Cryptography prior to the modern age was effectively synonymous with encryption, converting information from a readable state to unintelligible nonsense. The sender of an encrypted message shares the decoding technique only with intended recipients to preclude access from adversaries. The cryptography literature often uses the names Alice ("A") for the sender, Bob ("B") for the intended recipient, and Eve ("eavesdropper") for the adversary.^[5] Since the development of rotor cipher machines in World War I and the advent of computers in World War II, cryptography methods have become increasingly complex and its applications more varied.

Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in actual practice by any adversary. While it is theoretically possible to break into a well-designed system, it is infeasible in actual practice to do so. Such schemes, if well designed, are therefore termed "computationally secure"; theoretical advances, e.g., improvements in integer factorization algorithms, and faster computing technology require these designs to be continually reevaluated, and if necessary, adapted. There exist information-theoretically secure schemes that provably cannot be broken even with unlimited computing power, such as the one-time pad, but these schemes are much more difficult to use in practice than the best theoretically breakable but computationally secure schemes.

The growth of cryptographic technology has raised a number of legal issues in the information age. Cryptography's potential for use as a tool for espionage and sedition has led many governments to classify it as a weapon and to limit or even prohibit its use and export.^[6] In some jurisdictions where the use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation.^[7][8] Cryptography also plays a major role in digital rights management and copyright infringement disputes in regard to digital media.^[9]

  Terminolo

Alphabet shift ciphers are believed to have been used by Julius Caesar over 2,000 years ago.^[5] This is an example with k = 3. In other words, the letters in the alphabet are shifted three in one direction to encrypt and three in the other direction to decrypt.

The first use of the term cryptograph (as opposed to cryptogram) dates back to the 19th century—originating from The Gold-Bug, a story by Edgar Allan Poe.^{[10][11][broken footnote]}

Until modern times, cryptography referred almost exclusively to encryption, which is the process of converting ordinary information (called plaintext) into unintelligible form (called ciphertext).^[12] Decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext. A cipher (or cypher) is a pair of algorithms that carry out the encryption and the reversing decryption. The detailed operation of a cipher is controlled both by the algorithm and, in each instance, by a "key". The key is a secret (ideally known only to the communicants), usually a string of characters (ideally short so it can be remembered by the user), which is needed to decrypt the ciphertext. In formal mathematical terms, a "cryptosystem" is the ordered list of elements of finite possible plaintexts, finite possible cyphertexts, finite possible keys, and the encryption and decryption algorithms which correspond to each key. Keys are important both formally and in actual practice, as ciphers without variable keys can be trivially broken with only the knowledge of the cipher used and are therefore useless (or even counter-productive) for most purposes.

Historically, ciphers were often used directly for encryption or decryption without additional procedures such as authentication or integrity checks. There are, generally, two kinds of cryptosystems: symmetric and asymmetric. In symmetric systems, the only ones known until the 1970s, the same key (the secret key) is used to encrypt and decrypt a message. Data manipulation in symmetric systems is faster than asymmetric systems in part because they generally use shorter key lengths. Asymmetric systems use a "public key" to encrypt a message and a related "private key" to decrypt it. The use of asymmetric systems enhances the security of communication, largely because the relation between the two keys is very hard to discover.^[13] Examples of asymmetric systems include RSA (Rivest–Shamir–Adleman), and ECC (Elliptic Curve Cryptography). Quality symmetric algorithms include the commonly used AES (Advanced Encryption Standard) which replaced the older DES (Data Encryption Standard).^[14] Not very high quality symmetric algorithms include the assorted children's language tangling schemes such as Pig Latin or other cant, and indeed effectively all cryptographic schemes, however seriously intended, from any source prior to the invention of the one-time pad early in the 20th century.

In colloquial use, the term "code" is often used to mean any method of encryption or concealment of meaning. However, in cryptography, code has a more specific meaning: the replacement of a unit of plaintext (i.e., a meaningful word or phrase) with a code word (for example, "wallaby" replaces "attack at dawn"). A cypher, in contrast, is a scheme for changing or substituting an element below such a level (a letter, or a syllable or a pair of letters or ...) in order to produce a cyphertext.

Cryptanalysis is the term used for the study of methods for obtaining the meaning of encrypted information without access to the key normally required to do so; i.e., it is the study of how to "crack" encryption algorithms or their implementations.

Some use the terms cryptography and cryptology interchangeably in English, while others (including US military practice generally) use cryptography to refer specifically to the use and practice of cryptographic techniques and cryptology to refer to the combined study of cryptography and cryptanalysis.^[15][16] English is more flexible than several other languages in which cryptology (done by cryptologists) is always used in the second sense above. RFC 2828 advises that steganography is sometimes included in cryptology.^[17]

The study of characteristics of languages that have some application in cryptography or cryptology (e.g. frequency data, letter combinations, universal patterns, etc.) is called cryptolinguistics.

History of cryptography and cryptanalysis

Before the modern era, cryptography focused on message confidentiality (i.e., encryption)—conversion of messages from a comprehensible form into an incomprehensible one and back again at the other end, rendering it unreadable by interceptors or eavesdroppers without secret knowledge (namely the key needed for decryption of that message). Encryption attempted to ensure secrecy in communications, such as those of spies, military leaders, and diplomats. In recent decades, the field has expanded beyond confidentiality concerns to include techniques for message integrity checking, sender/receiver identity authentication, digital signatures, interactive proofs and secure computation, among others.

Classic cryptography

Reconstructed ancient Greek scytale, an early cipher device

The main classical cipher types are transposition ciphers, which rearrange the order of letters in a message (e.g., 'hello world' becomes 'ehlol owrdl' in a trivially simple rearrangement scheme), and substitution ciphers, which systematically replace letters or groups of letters with other letters or groups of letters (e.g., 'fly at once' becomes 'gmz bu podf' by replacing each letter with the one following it in the Latin alphabet). Simple versions of either have never offered much confidentiality from enterprising opponents. An early substitution cipher was the Caesar cipher, in which each letter in the plaintext was replaced by a letter some fixed number of positions further down the alphabet. Suetonius reports that Julius Caesar used it with a shift of three to communicate with his generals. Atbash is an example of an early Hebrew cipher. The earliest known use of cryptography is some carved ciphertext on stone in Egypt (ca 1900 BCE), but this may have been done for the amusement of literate observers rather than as a way of concealing information.

The Greeks of Classical times are said to have known of ciphers (e.g., the scytale transposition cipher claimed to have been used by the Spartan military).^[18] Steganography (i.e., hiding even the existence of a message so as to keep it confidential) was also first developed in ancient times. An early example, from Herodotus, was a message tattooed on a slave's shaved head and concealed under the regrown hair.^[12] More modern examples of steganography include the use of invisible ink, microdots, and digital watermarks to conceal information.

In India, the 2000-year-old Kamasutra of Vātsyāyana speaks of two different kinds of ciphers called Kautiliyam and Mulavediya. In the Kautiliyam, the cipher letter substitutions are based on phonetic relations, such as vowels becoming consonants. In the Mulavediya, the cipher alphabet consists of pairing letters and using the reciprocal ones.^[12]

In Sassanid Persia, there were two secret scripts, according to the Muslim author Ibn al-Nadim: the šāh-dabīrīya (literally "King's script") which was used for official correspondence, and the rāz-saharīya which was used to communicate secret messages with other countries.^[19]

David Kahn notes in The Codebreakers that modern cryptology originated among the Arabs, the first people to systematically document cryptanalytic methods.^[20] Al-Khalil (717–786) wrote the Book of Cryptographic Messages, which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels.^[21]

First page of a book by Al-Kindi which discusses encryption of messages

Ciphertexts produced by a classical cipher (and some modern ciphers) will reveal statistical information about the plaintext, and that information can often be used to break the cipher. After the discovery of frequency analysis, by the Arab mathematician and polymath Al-Kindi (also known as Alkindus) in the 9th century,^[22][23][24] nearly all such ciphers could be broken by an informed attacker. Such classical ciphers still enjoy popularity today, though mostly as puzzles. Al-Kindi wrote a book on cryptography entitled Risalah fi Istikhraj al-Mu'amma (Manuscript for the Deciphering Cryptographic Messages), which described the first known use of frequency analysis and cryptanalysis techniques. An important contribution of Ibn Adlan (1187–1268) was on sample size for use of frequency analysis.

16th-century book-shaped French cipher machine, with arms of Henri II of France

Enciphered letter from Gabriel de Luetz d'Aramon, French Ambassador to the Ottoman Empire, after 1546, with partial decipherment

Language letter frequencies may offer little help for some extended historical encryption techniques such as homophonic cipher that tend to flatten the frequency distribution. For those ciphers, language letter group (or n-gram) frequencies may provide an attack.

Essentially all ciphers remained vulnerable to cryptanalysis using the frequency analysis technique until the development of the polyalphabetic cipher. While it was known to Al-Kindi to some extent, it was first clearly described in the work of Al-Qalqashandi (1355–1418), based on the earlier work of Ibn al-Durayhim (1312–1359), describing a polyalphabetic cipher in which each plaintext letter is assigned more than one substitute. It was later also described by Leon Battista Alberti around the year 1467, though there is some indication that Alberti's method was to use different ciphers (i.e., substitution alphabets) for various parts of a message (perhaps for each successive plaintext letter at the limit). He also invented what was probably the first automatic cipher device, a wheel that implemented a partial realization of his invention. In the Vigenère cipher, a polyalphabetic cipher, encryption uses a key word, which controls letter substitution depending on which letter of the key word is used. In the mid-19th century Charles Babbage showed that the Vigenère cipher was vulnerable to Kasiski examination, but this was first published about ten years later by Friedrich Kasiski.^[28]

Although frequency analysis can be a powerful and general technique against many ciphers, encryption has still often been effective in practice, as many a would-be cryptanalyst was unaware of the technique. Breaking a message without using frequency analysis essentially required knowledge of the cipher used and perhaps of the key involved, thus making espionage, bribery, burglary, defection, etc., more attractive approaches to the cryptanalytically uninformed. It was finally explicitly recognized in the 19th century that secrecy of a cipher's algorithm is not a sensible nor practical safeguard of message security; in fact, it was further realized that any adequate cryptographic scheme (including ciphers) should remain secure even if the adversary fully understands the cipher algorithm itself. Security of the key used should alone be sufficient for a good cipher to maintain confidentiality under an attack. This fundamental principle was first explicitly stated in 1883 by Auguste Kerckhoffs and is generally called Kerckhoffs's Principle; alternatively and more bluntly, it was restated by Claude Shannon, the inventor of information theory and the fundamentals of theoretical cryptography, as Shannon's Maxim—'the enemy knows the system'.

Different physical devices and aids have been used to assist with ciphers. One of the earliest may have been the scytale of ancient Greece, a rod supposedly used by the Spartans as an aid for a transposition cipher. In medieval times, other aids were invented such as the cipher grille, which was also used for a kind of steganography. With the invention of polyalphabetic ciphers came more sophisticated aids such as Alberti's own cipher disk, Johannes Trithemius' tabula recta scheme, and Thomas Jefferson's wheel cypher (not publicly known, and reinvented independently by Bazeries around 1900). Many mechanical encryption/decryption devices were invented early in the 20th century, and several patented, among them rotor machines—famously including the Enigma machine used by the German government and military from the late 1920s and during World War The ciphers implemented by better quality examples of these machine designs brought about a substantial increase in cryptanalytic difficulty after WWI.