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Title, Concepts of Information Theory and Coding. Author, P. S. Satyanarayana. Publisher, MedTech, ISBN, , Length, CONCEPTS OF INFORMATION THEORY & CODING. Front Cover. P. S. Satyanarayana. Scientific International, – pages. Concepts of Information Theory & Coding [Paperback]. ISBN: Edition: 1st Edition. Author: P.S. Satyanarayana. Year: Pages:

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Skip to main satyaanarayana. Log In Sign Up. This book contains research papers, invited talks, abstracts, presentatins. Dr Satyanarayana Bhavanari M. Scientist AwardeeFellow – A. Mohiddin Shaw Shaik M. Dr Bhavanari Satyanarayana, Mr. Inaugural Address 02 Prof. Praveena Efficient using graphs in St.

CONCEPTS OF INFORMATION THEORY & CODING. – P. S. Satyanarayana – Google Books

Latha Efficient using graphs in St. Nagendram A Note on Inversive L. University and Hamiltonian Circuit 23 Mr. Sreenivasulu and an Interval Graphs V. Rama Latha 25 Bhavanari Mallikarjun Nanotechnology: A Paris newspaper, dated may 31stcarried the following news: At point blank theoy, each of them was given a pistol and fired. Only one theoru the pistols was loaded. Although his fame as a revolutionary was transient, his mathematics informayion timeless.

Galois Theory and Galois Groups are common currency in mathematics to day. As an young man of twenty he joined the ranks of the galaxy of immortals in mathematics. He was Evariste Galois. Inspite of lackluster career at the school, as he failed twice in the entrance examination to the prestigious educational institution, the Ecole Polytechnique, the young Galois devoured the advance mathematical texts of Legendre and Lagrange. When he was just 17 years old, he began making fundamental discoveries in the theory of polynomial equations.

By looking at sataynarayana series of mishaps in the short, but eventful life of Galois, it is amazing that the mathematical document that he wrote under the shadow of death the following morning, on the fateful night of May yheorydid not go to oblivion, but survived for the benefit of posterity. Galois was a radical republican during the turbulent days, after the fall of Napoleon in France.

He was something of a romantic figure in the history of mathematics. For his rebellious and revolutionary activities he was arrested and served jail terms. Soon after his release from the prison he was to stay in a place to avoid the risk of contacting the cholera epidemic. At that point of time he fell desperately in love with a girl which led to the tragic end of his life. While hteory teens Galois was attracted towards solving the quantic equation in Algebra.

To understand and appreciate the achievement of Galois, the romantic and tragic mathematician, we need to review the state of the theory of equations in Algebra, at his time. We find ample evidence from the history of mathematics of all the ancient civilizations, to confirm this fact.

Babylonians and Egyptians started elementary work in the theory of equations when they dealt with simple linear and quadratic equations involving only positive roots. Then entered Euclid and Archimedes with their geometric approaches to the problems. Thereafter Diophantus and Alkhwarizmi contributed in a big way and enriched the field.

Then followed the works of Indian mathematics, anmathematicians: At the dawn of second millennium Omar Khayyam, Fibonacci and a few others from Europe and Middle East, worked in this area. Li Zhi also known as Li Ye of China did solve higher degree equations by numerical methods.

But till the times of Renaissance, the cubic and the quartic, defied the attempts of mathematicians in producing a universal solution in terms of radicals. When we can find solutions for a polynomial equation with rational coefficients using only rational numbers and the operations of addition, subtraction, division, multiplication and extraction of roots, we say that equation is soluble by radicals.


In the sixteenth century the Italian mathematicians, FerroTartagliaCardanoand Ludovico Ferrarisucceeded in breaking the impasse by their ingenious methods. In the subsequent centuries, mathematicians broke their heads to extend the methods of the Renaissance scholars, but their attempts proved futile. Mathematicians of repute like: Thus numerous mathematicians tried their hand at this, producing valuable insights into the deep structure of equations and offering several new methods for solving equations of degrees three and four.

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Others produced practical numerical methods for solving equations of degrees higher than four, using successive approximations. They expanded and enriched the field of the theory of equations enormously. But the sole credit of hitting the nail on its head goes to the young Galois who gave the final verdict on the polynomial equation that could not be solved in the most general way.

He had the advantage of the availability of the works of Lagrange and Abel to make ground breaking advances which made him immortal in the world of modern mathematics. Originally Galois used permutation groups satyanagayana describe how the various roots of a given polynomial equation are related to each other.

In fact Group Theory is the discovery of Galois. It originated in the study of symmetric functions – the coefficients of a polynomial with sign attached are the elementary symmetric polynomials in the roots.

From this view point the Discriminant of a quadratic polynomial is a symmetric function in the roots which reflects properties of the roots. While Ruffin and Abel established that the general quantic could not be solved by radicalshowever certain particular quinticscan be solvedand the precise criterion by which a given quantic or higher polynomial could informatipn determined to be solvable or not was given by Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots — in modern termsits Galois group had a certain structure that is, in other words, whether or not it was a solvable group?

This group is always solvable for polynomials of degree four or less. This explains why there is no general formulaic solution for the quantic polynomial and polynomials of higher degrees.

How can permutations of oc tell us anything about solvability? The fact that permutations may provide at least some new and meaningful information had long been known in the nonmathematical world. When Galois was asked about the solvability of an equation, he did not ask simply whether an equation could be solved or not, but rather what kind of a solution it might have. Finally what has Galois done? He had not found a new solution in the manner of og Renaissance mathematicians of Italy.

He discovered a fundamental truth about equations of different degrees in the way Lagrange did and proved this truth, as Abel has done. What ckncepts had found was a method to determine whether any particular equation was solvable by radicals and that method was next to useless from practical point of view. And perhaps it is no wonder that the young man who styled himself as a romantic hero, who mused about the proximity of death and sacrificed his life for love and liberty, would always value beauty over practicality.

The future of mathematics according to Galois, lay not in producing specific results and resolving specific problems. The mathematicians sagyanarayana future, Galois believed, will be concerned only with the ttheory inner relations that govern the mathematical world itself, paying no heed to the messy and untidy demands of lengthy and satyanarayzna computations.


We quote Matthew Pordage: What satyanarauana profound truth lies hidden in the words of Matthew Pordage!

Galois indeed inspired and paved the path toward the type of mathematics cherished by the Bourbakians, a century later: Earlier to Galois, equations were always classified only by their degree: Galois discovered that symmetry was a more important characteristic.

The Galois group of an equation is the largest group of permutations the solutions or roots that leaves the values of certain combinations of these solutions unchanged. The maximum number of possible permutations of n solutions of a polynomial equation of degree n is n!. The group containing all of these permutations is the group we usually represent as Sn.

Galois was able to show that for any degree n, we can always find equations for which Galois group is the full Sn. There are fifth degree equations, for instance, for which the Galois group is S5.

We say that such equations possess the maximum symmetry possible. Though he did not use the word normal subgroup essentially the concept was his in his own style in its formative days. To be brief, he demonstrated that general polynomial equation of degree n could be solved by radicals if and only if every subgroup N of the group of permutations Sn is a normal subgroup. This demonstrated the impossibility of finding a solution in radicals for all general polynomial equations of degree greater than 4.

What does it take for an equation to be solvable by a formula? Galois showed that to be delighted with the luxury of having a formulaic solution, equations must have a Galois group of a very particular type. Galois was excited to call it a solvable group. When the Galois group of an equation is solvable, the process of the solution of the equation can be broken up into simpler steps, each involving only the solutions of equations of lower degree.

To sum up we list three criteria in the scheme of Galois. Thus a family of problems dating back for centuries, concerning the solution of polynomial equations, has been wrapped up once for all.

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Goldie, University of Leedsx Introduction: It is well known that the dimension of a vector space is defined as the number of elements in its satyanaryana. One can define a basis of a vector space as a maximal set of linearly independent vectors or a minimal set of vectors, which span the space. The former, when generalized to modules over rings, becomes the concept of Goldie Dimension. We discuss some results and examples related to the dimension in Vector Spaces as well as Modules over Rings.

Elementary Concepts in Vector Conceppts 1. We use F for field. The elements of F are called scalars and the elements of V are called vectors. Then i f is a group homomorphism or group endomorphism. Then K is a vector space over F. Then Fn is a vector space. Consider F[x], the ring of polynomials over F. Then Vn is a vector space over F.

Then W is called a subspace of V if W is a vector space over F under the same operation.