Monday, December 17, 2012

Year 2012-National mathematical Year


Relevance of National Mathematical Year

To encourage and facilitate the study of mathematics as an academic discipline in the country, the Prime Minister Manmohan Singh declared the year 2012 as a ‘National Mathematical Year’ as a tribute to maths wizard Srinivasa Ramanujan and also declared December 22, the birthday of Ramanujan, as 'National Mathematics Day'. Prime Minister expressed concern over dwindling interest in mathematics and subject experts and emphasized on mathematical research.

Mathematics in some form or other has been used since the early age of human civilization. But its use in today’s world has assumed great importance, since without its application higher technology cannot be mastered and harnessed for increasing production of goods and services and promoting human welfare. Technology generally follows the advancement of science which is also vitally dependent on the use of Mathematics. Over the centuries there has been spectacular progress in the development of Mathematics as a branch of knowledge. And without the application of Mathematics on a wide scale no country can march forward in line with the general progress of human knowledge and thought. Therefore learning of Mathematics and promoting the horizon of knowledge by advanced researches in Mathematics should be over emphasized.The importance of Mathematics in India can be seen by a well known verse in Sanskrit of ‘Vedang Jyotish’  (written 1000 BC) as:

यथा शिखा मयूराणां , नागानां मणयो  यथा तद्वद् वेदाङ्ग शास्त्राणां, गणितं मुर्धिनं वर्तते   

"As the crown on the head of a peacock, and as the gem on the hood of a snake,so stands Mathematics crowned above all disciplines of knowledge." This fact was well known to intellectuals of India that is why they gave special importance to the development of Mathematics, right from the beginning. Indian mathematicians made great strides in developing arithmetic, algebra, geometry, infinite series expansions and calculus. The oldest evidence of mathematical knowledge to Indians is being found in Indus Valley Civilization. The metallic seals found in the excavations of Mohan-Jo-Daro and Harappa indicates that the people of this civilization had the knowledge of numbers. It is also clear from the pottery and other archaeological remains that they had the knowledge of measurement and Geometry even in crude form. Indian works, through a variety of translations, have had significant influence throughout the world.

But unfortunately our students do not know even about the famous names of Indian mathematics. They are unaware about the contribution of Āryabhata in place value number system, the geometrical concepts given by Brahmagupta , the fundamental work of Bhaskarachrya on arithmetic, the currently used method of calculating the Least Common Multiple (LCM) of given numbers in Ganita-sāra-sańgraha of Mahāvīrācārya, Ramanujan’s extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions.

The present mathematical knowledge has not dropped as a bolt from the blue, nor a product of some magical tricks. The apparently ready-made knowledge and results have been achieved after centuries of efforts, often painful, by hundreds of Mathematicians and Historians through the ages. Lot of discoveries and inventers contributed to the fruits, facilities and luxuries which we enjoy today were the contribution of Indian mathematicians. From the notion of zero to the modern concept of computational number theory, cryptography and fuzzy set theory, their contribution is significant.

Kim Plofker from Netherlands says that “Indian mathematical science is extremely important and has a significant effect on the world’s knowledge as it is today. The lack of available resources has kept us under informed about the developments that have taken place in India.” It is the need of the hour to carry forward the legacy of great mathematicians so as to encourage and nurture the glorious tradition of the country in mathematics. 

The country needs to produce more mathematicians of the caliber of Ramanujan to become a knowledge superpower in the world.In the published curriculum development report of University Grant Commission in 1989, it was suggested that History of Mathematics should be a part of curriculum to make the subject interesting. But nothing has been done in this reference. I strongly believe that student must know the process and development of mathematical thinking, formulae and its principles. If one knows the process of developing mathematical model, can not be frustrated at any stage.

Wednesday, July 13, 2011

Trilokya Dīpaka of Pt. Vāmadeo (14th C. A.D.)

Trailokya Dîpaka: A mathematical treatise containing the detailed structure of the universe.

"Lokānuyoga Manuscript" by Jinasena (9th C.A.D.)

Lokānuyoga -A Mathematical Manuscript of  Jinasena (9th C.A.D.) on Karmic theory. 

Mathematical Manuscript "Jyotirjñānavidhi" by Śrīdharācārya (8th c. A.D.)

The manuscript dealt with trigonometry, combinatorics, fractions etc. and also provided mathematical formulae for a variety of practical problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns.

Monday, May 16, 2011

Life is real, don’t make it as complex: A mathematical Approach

The main problem of our life is ‘ego’.  If ego has gone from us then 

everything else will go away from us automatically.

The present day life is complex, not because nature gives us a complex life but because of our ego (I) we make the real life as a complex one. Now, we compare this with complex number a+ib, where a and b are real numbers. Further a is called real part, b is called imaginary part and i = sqrt (-1),  this number a+ib becomes complex because ‘i’ is there between a and b, otherwise it is sum of two real numbers a and b which is a real number.

Now we will discuss how to make this complex number (complex life) as a real number (real life)? This can be explained in the following easy ways:

· If we remove ‘I’ (‘I’ ness from us, that is, the ego from us) then the number (life) becomes real.

 Instead of living in the present, that is, in reality, we live most of the time either in the past or in the future. So we are always in the imaginary world. This makes our life as complex. Now by allowing the imaginary part of our life tends to zero (b→0) we can make the complex life (number) as a real life (number). This shows that we have to live in the present and utilize it as effectively as possible.

·        We think all the time about ourselves forgetting the absolute (God), which is the reason for our complexity. If we are in the remembrance of God then life becomes real. It is like taking absolute value of a complex number, that is,                      a+ib=sqrt (a^2 + b^2), which is a real number. From this can understand that if we are in the remembrance of God then the life becomes real.

·         Finally, if we find the root (origin) of positive value then we get real value. But if we try to find he root of negative value we get complex value, i.e., sqrt (-x^2)= ix. So in our life if we have any negative characters then there is no need to find the root of it, just remove it, so that life becomes real.


Spirituality says that we suffer because of our desire. Buddha said cut the desire of chain anywhere then in will fall. So we can define happiness as follows:

Happiness = b/a, where a = Number of desires we have, b = Number of desires fulfilled. 

The fraction b/a value will increase by decreasing the value of a, that is we will be more happy if we reduce the number of desires. The fraction value will decrease by increasing the value of a, that is we will be less happy if we increase the number of desires. One who has infinite desires will not be happy at all. On the other hand one who has no desire happy at all times. The saints and children are the only people who allow a→0, so they are always happy.

From: Values Through Mathematics

Thursday, February 24, 2011

Education is not Preparation for life; Education is life itself

Logical reasoning and deductions, understanding of relationships, finding implications, solving problematic situations, application of principles, Interpretation of data, Critical analysis, Evaluation and Discrimination are the challenges to the brain storming sessions. The real goal of education is making an interdependent global society. A society where enable and disable, superior and inferior successful and failure, winner and loser –all understand and learn from each other’s. It’s not just about the higher class, or highly qualified or even enables.  It’s about each and every individual. Learning is a place where paradise can be created. The classroom with all its limitations remains a location of possibility. This is education as the practice of freedom. The awakening of intelligence is more important than the cultivation of memory, both in life and in academics. The greatest things in life are those that cannot be taught but they can be learnt. The feeling of love, respect, beauty, and friendship, cannot be taught but like sensitivity, it can be awakened and this is an essential part of intelligence. The aims of education may vary a little from country to country but essentially, all over the world, education is aiming to produce a human being who is intelligent, knowledgeable, hard working, efficient, disciplined, smart, and successful; hopefully a leader in his field of endeavor. 

Tuesday, February 15, 2011

English Rendezvous with Math

It is indeed a myth that math is not about language but only about symbols and numbers. English language proficiency is strongly correlated to mathematical problem-solving skills. Promoting English language in teaching Mathematics is a positive attitude towards the acceptance of fascination of Mathematics. English is a practical and accessible language. It covers a wide range of getting sources and information of study material, new researches, subject knowledge and new technologies. Creating English Learning Environment in teaching Math is a need of developing countries for their development.

English language provides numerous opportunities for both students and teachers to engage in mathematical experiences within international learning communities linked via computer networking facilities. Everyone should develop his or her English language proficiency to understand the role of mathematical sciences in today’s world.

No doubt English-Mathematics combination is a highly intellectual one in which all the disciplines intersect. Mathematical culture and English learning upbringing provide a bridge for entering into intellectual galaxies.

Globalization is leaving its impact on Higher Education by way of new enhancement in technology, communication and International research work. In this era, we cannot imagine the development of any country without accepting the knowledge of English language as a valuable resource.

Mathematics is a highly intellectual discipline. The world of Mathematics is essentially a man-made world and has been designed to meet man’s ideas of intellectual perfection. This fascination should not be limited to its cultivators, the mathematicians or its users, the scientists, but must be felt by every citizen. 

Young people need to think innovatively and creatively on challenging problems from all fields of life by making mathematical models, drawing conclusions, comparing them with observations at local and global level. Of course it is true that Mathematics needs cognitive and logical thinking and the language part is not much essential while teaching but at the same time, the teaching of Mathematics in English provides the challenge to each others’ learning and practices and generally to increase their knowledge about problems of Mathematics around the world and about the solutions that have worked in other places and it is regarded as one of the corner stones for the sustaining of young ones.

Teaching Mathematics in English provides many opportunities: More accessibility like creating competitive environment, finding Global outreach, getting more accessibility of study material etc. So it is a need of any country to motivate and promote Mathematics teaching in English so that students satisfy the needs of the Mathematical Community.

Tuesday, February 1, 2011

World Maths Day

This year World Maths Day is going to be celebrate on 1st March. You can register for this through  .  

Saturday, January 1, 2011

Best wishes-New Year 2011

Wish you all a very happy and prosperous new year.........2011.
Take some resolutions to get the success ahead. All the best!!!!!!!!!

Friday, November 19, 2010

Differential Equations (Part 1)

You can get the knowledge of Ordinary differential equations and its types through the link here

Sunday, November 7, 2010

Set Theory Notations

Set Presentation

Common notations

• Z=the set of integers = {…-3, -2, -1, 0, 1, 2, 3,…..}

• N=the set of natural numbers or nonnegative integers 

• Z+=the set of positive integers

• Q=the set of rational numbers={a/b a,b is integer, b not zero}

• Q+=the set of positive rational numbers

• Q*=the set of nonzero rational numbers

• R=the set of real numbers

• R+=the set of positive real numbers

• R*=the set of nonzero real numbers

• C=the set of complex numbers

Set Presentation

Thursday, November 4, 2010

Monday, September 13, 2010

Introduction to Operation Research

The course " Computer Oriented Optimization Technique" is associated with the term Operation Research.

What is Operation Research?

  • Operations research, operational research, or simply O.R., is the use of mathematical models, statistics and algorithms to aid in decision-making. It is the discipline of applying appropriate analytical methods to help make better decisions.
  • It is most often used to analyze complex real-world systems, typically with the goal of improving or optimizing performance.
  • OR is an interdisciplinary branch of applied mathematics and formal science that uses methods such as:
                     Mathematical modeling




• The modern field of operations research arose during World War II.

• Scientists in the United Kingdom including Patrick Blackett, Cecil Gordon, C. H. Waddington, Owen Wansbrough-Jones and Frank Yates, and in the United States with George Dantzig looked for ways to make better decisions in such areas as logistics and training schedules. After the war it began to be applied to similar problems in industry.

• It is known as "operational research" in the United Kingdom (and "operational analysis" within the UK military and UK Ministry of Defence, where OR stands for "Operational Requirement") and as "operations research" in most other English-speaking countries, but "OR" is the common abbreviation everywhere. With expanded techniques and growing awareness, OR is no longer limited to only operations, and the introduction of computer data collection and processing has relieved analysts of much of the more mundane labor.


• Statistics

• Optimization

• Probability theory

• Queing theory

• Game theory

• Graph theory

• Decision analysis and

• Simulation

Because of the computational nature of these fields, OR also has ties to computer science.

Scope of OR

Operations research plays an increasingly important role in both the public and private sectors. Operations research addresses a wide variety of issues in transportation, inventory planning, production planning, crew planning, communication network design and operation, computer operations, financial assets and risk management, revenue management, market clearing and many other topics that aim to improve business productivity. In the public domain it deals with such topics as energy policy, defense, health care, water resource planning, forestry management, design and operation of urban emergency systems and criminal justice.

Operation Research and Decision Making

Operation research or management science as the name suggests, is the science of managing. As is known, management is most of the time making decisions. It is thus decision science which helps management to make better decision. Decision is , in fact, a pivotal world in managing. It is not only the headache of management , rather all of us make decision. We daily decide about minor to major issues. We choose to be engineers , doctors, lawyers , managers etc.

Decision making can be improved and , in fact , there is a scope of large scale improvement. The essential characteristics of all decision are

1. Objectives 2. Alternatives 3. Influencing factors

Scope of OR in Management

Operations research is problem solving and decision making science. It is a kit of scientific programmable rules providing the management a ‘quantitative basis’ for decisions regarding the operation under its control. Some of the areas of management where OR techniques have been successfully applied are:

Allocation and distribution

Optimal allocation of limited resources such as men, machine, materials, time and money. Location and size of whearhouse, distribution centres, retail depots etc.

Production and facility planning

• Selection ,location and design of production plants.

• Project scheduling and allocation of resources.

• Maintenance policy.

• Determination of the number and size of the items to be produced.


• What , how and when to purchase at the minimum procurement cost?

• Bidding and replacement policies.


• Product selection, trimming and competitive actions.

• Selection of advertising media.

• Demand forecasts and stock levels.

• Customer preference for size, color and packing of various products.


Operation Research has wide scope and has been successfully applied in the following areas of Financial Management: Cash Management, Inventory control , Simulation Technique, Capital Budgeting.

Cash Management :

A financial manager of responsible for adequate supply of funds to all the sections, departments and units of the organization as adequate funds are essential for their proper function throughout the year. Linear programming techniques are helpful to determine the allocation of funds to each section. Linear programming techniques have also been applied to identify sections having excess funds; these funds may be diverted to section that need them.

Inventory control:

In big organizations the amount invested in inventories can run into millions of rupees. Inventory control techniques of OR can help management to develop better inventory policies and bring down the investment in inventories. These techniques help to achieve optimum balance between inventory carrying costs , ordering costs and shortage costs. They help to determine which items to hold , how much to hold when order and how much to order.

Simulation Technique :

Simulation consider various factors that affect present and projected cost of borrowing money from commercial banks, and tax rates etc. and provides an optimum combination of financing for the desired amount of capital. Simulation replaces subjective estimates , judgment and hunches of the management by providing reliable information.

Capital Budgeting :

It involves of various investment proposals (market introduction of a new product or replacement of an equipment by a new one)


• Selection of personal, determination of retirement age and skills.

• Recruitment policies and assignment of jobs.

Research and development:

1. Determination of areas for research and development.

2. Reliability and control of development projects.

3. Selection of projects and preparation of their budgets.

From all the above areas of application, it can be concluded that OR can be widely used in taking timely management decision and can be also used as a corrective measure.

 OR Models

The operations research analyst has a wide variety of methods available for problem solving. For mathematical programming models there are optimization techniques appropriate for almost every type of problem, although some problems may be difficult to solve. For models that incorporate statistical variability there are methods such as probability analysis and simulation that estimate statistics for output parameters. In most cases the methods are implemented in computer programs.

Linear Programming

• Linear programming is a widely used model type that can solve decision problems with many thousands of variables.

• Generally, the feasible values of the decisions are delimited by a set of constraints that are described by mathematical functions of the decision variables.

• The feasible decisions are compared using an objective function that depends on the decision variables.

• For a linear program the objective function and constraints are required to be linearly related to the variables of the problem.

Network Flow Programming

• The term network flow program describes a type of model that is a special case of the more general linear program. The class of network flow programs includes such problems as the transportation problem, the assignment problem, the shortest path problem, the maximum flow problem, the pure minimum cost flow problem, and the generalized minimum cost flow problem.

• It is an important class because many aspects of actual situations are readily recognized as networks and the representation of the model is much more compact than the general linear program.

• When a situation can be entirely modeled as a network, very efficient algorithms exist for the solution of the optimization problem, many times more efficient than linear programming in the utilization of computer time and space resources.

Network models are constructed by the Math Programming add-in and may be solved by either the Excel Solver, Jensen LP/IP Solver or the Jensen Network Solver.

Integer Programming

Integer programming is concerned with optimization problems in which some of the variables are required to take on discrete values. Rather than allow a variable to assume all real values in a given range, only predetermined discrete values within the range are permitted. In most cases, these values are the integers, giving rise to the name of this class of models.

Nonlinear Programming

When expressions defining the objective function or constraints of an optimization model are not linear, one has a nonlinear programming model. Again, the class of situations appropriate for nonlinear programming is much larger than the class for linear programming. Indeed it can be argued that all linear expressions are really approximations for nonlinear ones.

Dynamic Programming

Dynamic programming (DP) models are represented in a different way than other mathematical programming models. Rather than an objective function and constraints, a DP model describes a process in terms of states, decisions, transitions and returns. The process begins in some initial state where a decision is made. The decision causes a transition to a new state. Based on the starting state, ending state and decision a return is realized. The process continues through a sequence of states until finally a final state is reached. The problem is to find the sequence that maximizes the total return.

Stochastic Programming

The mathematical programming models, such as linear programming, network flow programming and integer programming generally neglect the effects of uncertainty and assume that the results of decisions are predictable and deterministic. This abstraction of reality allows large and complex decision problems to be modeled and solved using powerful computational methods.

Combinatorial Optimization

The most general type of optimization problem and one that is applicable to most spreadsheet models is the combinatorial optimization problem. Many spreadsheet models contain variables and compute measures of effectiveness. The spreadsheet user often changes the variables in an unstructured way to look for the solution that obtains the greatest or least of the measure. In the words of OR, the analyst is searching for the solution that optimizes an objective function, the measure of effectiveness. Combinatorial optimization provides tools for automating the search for good solutions and can be of great value for spreadsheet applications.

Stochastic Processes

In many practical situations the attributes of a system randomly change over time. When aspects of the process are governed by probability theory, we have a stochastic process.

The model is described in part by enumerating the states in which the system can be found. The state is like a snapshot of the system at a point in time that describes the attributes of the system.

Discrete Time Markov Chains

Say a system is observed at regular intervals such as every day or every week. Then the stochastic process can be described by a matrix which gives the probabilities of moving to each state from every other state in one time interval. Assuming this matrix is unchanging with time, the process is called a Discrete Time Markov Chain (DTMC). Computational techniques are available to compute a variety of system measures that can be used to analyze and evaluate a DTMC model. This section illustrates how to construct a model of this type and the measures that are available.

Continuous Time Markov Chains

Here we consider a continuous time stochastic process in which the duration of all state changing activities are exponentially distributed. Time is a continuous parameter. The process satisfies the Monrovian property and is called a Continuous Time Markov Chain (CTMC). The process is entirely described by a matrix showing the rate of transition from each state to every other state. The rates are the parameters of the associated exponential distributions. The analytical results are very similar to those of a DTMC. The ATM example is continued with illustrations of the elements of the model and the statistical measures that can be obtained from it.


When a situation is affected by random variables it is often difficult to obtain closed form equations that can be used for evaluation. Simulation is a very general technique for estimating statistical measures of complex systems. A system is modeled as if the random variables were known. Then values for the variables are drawn randomly from their known probability distributions. Each replication gives one observation of the system response. By simulating a system in this fashion for many replications and recording the responses, one can compute statistics concerning the results. The statistics are used for evaluation and design.


• Applications in which operations research is currently used include:

• Critical path analysis or project planning: identifying those processes in a complex project which affect the overall duration of the project

• Designing the layout of a factory for efficient flow of materials

• Constructing a telecommunications network at low cost while still guaranteeing QOS (quality of service) or QOE (Quality of Experience) if particular connections become very busy or get damaged.

• Road traffic management and 'one way' street allocations i.e. allocation problems.

• Determining the routes of school buses (or city buses) so that as few buses are needed as possible

• Designing the layout of a computer chip to reduce manufacturing time (therefore reducing cost)

• Managing the flow of raw materials and products in a supply chain based on uncertain demand for the finished products.

• Efficient messaging and customer response tactics

• Robotic zing or automating human-driven operations processes

• Globalizing operations processes in order to take advantage of cheaper materials, labor, land or other productivity inputs

• Managing freight transportation and delivery systems (Examples: LTL Shipping, inter-modal freight transport)

• Scheduling:

• personnel staffing

• manufacturing steps

• project tasks

• network data traffic: these are known as queuing models or queuing systems.

• sports events and their television coverage

• Blending of raw materials in oil refineries

• Operations research is also used extensively in government where evidence-based policy is used.

Syllabus-Computer Oriented Optimization Techniques ( MCA 301)

Dear Students, Welcome to III sem!
Here is the syllabus for you of the course  MCA 301
Computer Oriented Optimization Techniques
Total Marks: 150 ( Theory Marks: 100, Sessional Marks: 50)

The entire course is divided into following 5 units:

Introduction of operation research. LP Formulations, Graphical method for solving LP’s with 2 variables, Simplex method, Duality theory in linear programming and applications, Integer linear programming, dual simplex method.

Transportation problem, Assignment problem.
Dynamic Programming : Basic Concepts, Bellman’s optimality principles, Dynamics programming approach in decision making problems, optimal subdivision problem.

Sequencing Models: Sequencing problem, Johnson’s Algorithm for processing n jobs through 2 machines, Algorithm for processing n jobs through 3 or more machines, Processing 2 jobs through n machines.

Project Management : PERT and CPM : Project management origin and use of PERT, origin and use of CPM, Applications of PERT and CPM, Project Network, Diagram representation, Critical path calculation by network analysis and critical path method (CPM), Determination of floats, Construction of time chart and resource labelling, Project cost curve and crashing in project management, Project Evaluation and review Technique (PERT).

Queuing Models : Essential features of queuing systems, operating characteristics of queuing system, probability distribution in queuing systems, classification of queuing models, solution of queuing M/M/1 :  /FCFS,M/M/1 : N/FCFS, M/M/S : /FCFS, M/M/S : N/FCFS.

Inventory Models : Introduction to the inventory problem, Deterministic Models, The classical EOQ (Economic Order Quantity) model, Inventory models with deterministe demands(no shortage & shortage allowed), Inventory models with probabilistic demand, multiitem determinise models.

Suggested Books:

1. Gillet B.E. : Introduction to Operation Research, Computer Oriented Algorithmic approach - Tata McGraw Hill Publising Co. Ltd. New Delhi.

2. P.K. Gupta & D.S. Hira, “Operations Research”, S.Chand & Co.

3. J.K. Sharma, “Operations Research: Theory and Applications”, Mac Millan.

4. S.D. Sharma, “Operations Research”, Kedar Nath Ram Nath, Meerut (UP).

5. S.S. Rao “Optimization Theory and Application”, Wesley Eastern.

6. Tata Hamdy, A “Operations Research - An Introduction”, Fifth Edition, Prentice Hall of India Pvt. Ltd., New Delhi.

7. Taha H.A. “Operations Research an Introduction” McMillan Publication.

Sunday, September 5, 2010

Happy Teachers Day

A central piece in Hindu scripture reads

"Gurur Brahma, Gurur Vishnu, Guru devo Maheshwaraha - Gurussaakshaath param brahma tasmai shree gurave namaha,"

 which translates as :
"The Guru is the Lord Brahma ,
   the Guru is the Lord Vishnu ,
   the Guru is the Lord Shiva ,
   the Guru is the Supreme Brahman         
   visible to our eyes.
   To that Guru we offer our salutations"
Guru means 'Teacher',
Lord Brahma means ' The Creator',
Lord Vishnu is 'The Preserver', 
Lord Shiva is 'The Destroyer',
and Supreme Brahaman is 'The Ultimate Reality'.   


Wednesday, September 1, 2010

Sunday, August 15, 2010

Happy Independence Day to all ............

On this great historical day I congratulate to all indians.

Tuesday, August 10, 2010

About Set theory

A set is an unordered well defined collection of distinct objects represented as a unit. The objects in a set are called its elements or members. By ‘well defined’ we mean that it must be possible to tell beyond doubt whether or not a given object belongs to the collection that we are considering. If we are asked to write the set of all intelligent students of a class, it is not possible to do so. No two persons will have the common list. Thus the collection of intelligent students in the class is not a set. It is not necessary that a set contains only the same type of objects. If we have a book, a football, a glass and a table, then these objects also form a set although they are different form each other.

George Cantor (1845-1918), John Venn (1834-1923), George Boole (1815-1864), Augustus De Morgan (1806-1871) influenced the development of set theory and logic.

George Cantor, in 1895, was the first to define a set formally. He is considered as the “Father of Set theory”. Cantor took the idea of set to a revolutionary level, unveiling its true power. By inventing a notion of size of set he was able compare different forms of infinity and, almost incidentally, to shortcut several traditional mathematical arguments. But the power of sets came at a price; it came with dangerous paradoxes. The paradoxes of set theory were a real threat to the security of the foundations. But with a lot of worry and care the paradoxes were sidestepped, first by Russell and Whitehead’s theory of stratified types and then more elegantly, in for example the influential work of Zermelo and Fraenkel. The notion of set is now a cornerstone of Mathematics.

Consider all of the men in a small town as members of a set. Now imagine that a barber puts up a sign in his shop that reads I shave all those men, and only those men, who do not shave themselves.

Obviously, we can further divide the set of men in this town into two further sets, those who shave themselves, and those who are shaved by the barber. To which set does the barber himself belong?

The barber cannot shave himself, because he has said he shaves only those men who do not shave themselves. Further, he can not shave himself because he shaves all men who do not shave themselves.

Russell’s Paradox arises within set theory by considering the set of all sets which are not members of themselves. Such a set appears to be a member of it if and only if it is not a member of itself.

The significance of Russell’s paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. In the eyes of many, it therefore appeared that no mathematical proof could be trusted once it was discovered that the logic and set theory apparently underlying all of mathematics was contradictory.

Set theory helps people to …

• see things in terms of systems

• organize things into groups

• begin to understand logic

In computer science, a set is an abstract data structure that can store certain values, without any particular order, and no repeated values. It is a computer implementation of the mathematical concept of a finite set.

In computer science, we are often concerned with sets of strings of symbols from some alphabet, for example the set of strings accepted by a particular automaton.

Types of Sets

Finite Sets

A set is called a finite set if we can count the number of elements of the set. In other words, a set of finite number of elements is called a finite set. Set of students in a school, set of people living in Indore, set of natural numbers from 20 to 50 are some of the examples of finite sets.

Infinite Sets

A set in which the number of elements cannot be count is called an infinite set. In other words, a set of infinite number of elements is called an infinite set. Set of all integers, set of points on a line and set of circles with a fixed centre are some of the examples of infinite sets.

Singleton set

A set which has only one member is known as a single member set or Singleton set.

For example, a set of natural numbers between 7 and 9 is a single member set having only one element 8 i.e.{8}.

Empty set or Null set

A set which contains no element is called an empty set or null set. It is also called it void set. In listing elements of an empty set we have braces { } and having nothing within the braces.

Null set is usually denoted by the Greek letter (phi). Clearly the set {0} is not an empty set as it is a set which has one element zero belonging to it. The set of triangles with two obtuse angles is { } and the set of natural numbers between 5 and 6 is also .

Empty set is a subset of every set.

Disjoint Sets

If no element of A is in B and no element of B is in A then A and B are called disjoint sets or we can say that if A∩B=ф then A and B are disjoint sets. For example, sets {1, 2, 5} and {2, 4, 6} are disjoint sets as there is no common element in them. In other words two sets which have no common members are called disjoint sets.

One-to One Correspondence Sets

Two sets are said to be in one- to- one correspondence if they can be matched in such a way that each element of one set is associated with a single element of the other.

Let A= {a, b, c} and B= {p, q, r, s} then it is clear that A and B are not in one-to-one correspondence. If the elements of A and B are matched then one element of B remains unmatched. Thus we say that A has fewer elements than B or B has more elements than A. In other words, the set B is larger than the set A.

Equivalent Sets

Two sets X and Y are said to be equivalent sets if the number of elements in X is equal to the number of elements in Y i.e., there is one-to-one correspondence between the elements of X and Y.

For example,{1. 2. 3} and {x, y, z} are equivalent sets. The symbol “~” is used to denote equivalence.

Thus A ~ B is read as ‘A is equivalent to B’.

Equal Sets

Two sets A and B are called equal sets if every element of A is also an element of B and every element of B is also an element of A. For example {r, s, t} and {t, r, s} are equal sets.

Obviously if two sets are equal, they are equivalent too but if two sets are equivalent they may not be equal.

For example, Given A= {2, 4, 6, 8} and B = {2, 8, 4, 6}. Here every element of A is a member of B and each element of B is also a member of A. Thus Set A= Set B.

Further, if X is a set of letters in the word ‘search’ and Y is a set of letters in the word ‘reaches’, then X= {s, e, a, r, c, h} and Y= {r, e, a, c, h, e, s}

X = Y

The order of elements and repetition of elements does not change a set.

Thursday, July 29, 2010

Role of Mathematics in other disciplines

A famous Jain Mathematician, Ācārya Mahāvira (9th century) writes that-“Bahubhirvi pralāpaih kim trailokye sacarācare, Yatkimcidvastu tatsarva ganitena binā nahi”. The verse shows the importance of Mathematics as-“What is good of saying much in vain? Whatever there is in all three worlds, which are possessed of moving and non-moving being all that indeed cannot exist as apart from Mathematics”.

Here are some main disciplines in which Role of Mathematics is widely accepted.

Physical Sciences

In mathematical physics, some basic axioms about mass, momentum, energy, force, temperature, heat etc. are abstracted, from observations and physical experiments and then the techniques of abstraction, generalization and logical deduction are used. It is the branch of mathematical analysis that emphasizes tools and techniques of particular use to physicists and engineers. It focuses on vector spaces, matrix algebra, differential equations, integral equations, integral transforms, infinite series, and complex variables. Its approach can be adapted to applications in electromagnetism, classical mechanics, and quantum mechanics.

In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics — the study of the motion of celestial objects. The role of Space Dynamics is very important in mechanics. Here we have to consider the trajectories which are time-optimal i.e. which take the least time in going from one point to another and in which the object starts and reaches the destination with zero velocity. Similarly we may have to consider energy- optimal trajectories. We have also to consider the internal and external ballistic of rocket and the path of inter continental ballistic missiles.

Fluid Dynamics

Understanding the conditions that result in avalanches, and developing ways to predict when they might occur, uses an area of mathematics called fluid mechanics. Many mathematicians and physicists applied the basic laws of Newton to obtain mathematical models for solid and fluid mechanics. This is one of the most widely applied areas of mathematics, and is also used in understanding volcanic eruptions, flight, ocean currents.

Civil and mechanical engineers still base their models on this work, and numerical analysis is one of their basic tools. In the 19th century, phenomena involving heat, electricity, and magnetism were successfully modeled; and in the 20th century, relativistic mechanics, quantum mechanics, and other theoretical constructs were created to extend and improve the applicability of earlier ideas. One of the most widespread numerical analysis techniques for working with such models involves approximating a complex, continuous surface, structure, or process by a finite number of simple elements, known as the finite element method (FEM). This technique was developed by the American engineer Harold Martin and others to help the Boeing Company analyze stress forces on new jet wing designs in the 1950s. FEM is widely used in stress analysis, heat transfer, fluid flow, and torsion analysis.
Fluid Dynamics is also very important in Atmospheric Sciences, in dynamic meteorology and weather prediction. Another use in the study of diffusion of pollutants in the atmosphere e.g. to find out what proportion of pollutants emitted from chimneys or refineries reach hospitals and other buildings. It is also needed for the study of effect of leakages of poisonous gases.

Computational Fluid Dynamics

Computational Fluid Dynamics is a discipline wherein we use computers to solve the Navier – Stokes equations for specified initial and boundary condition for subsonic, transonic and hypersonic flows. Many of our research workers use computers, but usually these are used at the final stage when drastic simplifications have already been made, partial differential equation have been reduced to ordinary differential equations and those equations have even been solved.

Physical Oceanography

Important fluid dynamics problem arise in physical oceanography. Problems of waves, tides, cyclones flows in bays and estuaries, the effect of efflux of pollutants from nuclear and other plants in sea water, particularly on fish population in the ocean are important for study. From defense point of view, the problem of under-water explosions, the flight of torpedoes in water, the sailing of ships and submarines are also important.


Math is extremely important in physical chemistry especially advanced topics such as quantum or statistical mechanics. Quantum relies heavily on group theory and linear algebra and requires knowledge of mathematical/physical topics such as Hilbert spaces and Hamiltonian operators. Statistical mechanics relies heavily on probability theory.

Other fields of chemistry also use a significant amount of math. For example, most modern IR and NMR spectroscopy machines use the Fourier transform to obtain spectra. Even biochemistry has important topics which rely heavily on math, such as binding theory and kinetics.
Even Pharmaceutical companies require teams of mathematicians to work on clinical data about the effectiveness or dangers of new drugs. Pure scientific research in chemistry and biology also needs mathematicians, particularly those with higher degrees in computer science, to help develop models of complicated processes.

Biological Sciences

Biomathematics is a rich fertile field with open, challenging and fascination problems in the areas of mathematical genetics, mathematical ecology, mathematical neuron- physiology, development of computer software for special biological and medical problems, mathematical theory of epidemics, use of mathematical programming and reliability theory in biosciences and mathematical problems in biomechanics, bioengineering and bioelectronics.

Mathematical and computational methods have been able to complement experimental structural biology by adding the motion to molecular structure. These techniques have been able to bring molecules to life in a most realistic manner, reproducing experimental data of a wide range of structural, energetic and kinetic properties. Mathematical models have played, and will continue to play, an important role in cellular biology. A major goal of cell biology is to understand the cascade of events that controls the response of cells to external legends (hormones, transport proteins, antigens, etc.). Mathematical modeling has also made an enormous impact on neuroscience. . Three-dimensional topology and two-dimensional differential geometry are two additional areas of mathematics when it interacts with biology. Its application is also very important to cellular and molecular biology in the area of structural biology. This area is at the interface of three disciplines: biology, mathematics and physics.

In Population Dynamics, we study deterministic and stochastic models for growth of population of micro-organisms and animals, subject to given laws of birth, death, immigration and emigration. The models are in terms of differential equations, difference equations, differential difference equations and integral equations.
In Internal physiological Fluid Dynamics, we study flows of blood and other fluids in the complicated network of cardiovascular and other systems. We also study the flow of oxygen through lung airways and arteries to individual cells of the human or animal body and the flow of synovial fluid in human joints. In External Physiological Fluid Dynamics we study the swimming of micro organisms and fish in water and the flight of birds in air.

In Mathematical Ecology, we study the prey predator models and models where species in geographical space are considered. Epidemic models for controlling epidemics in plants and animals are considered and the various mathematical models pest control is critically examined.

In Mathematical Genetics, we study the inheritance of genetic characteristics from generation to generation and the method for genetically improving plant and animal species. Decoding of the genetic code and research in genetic engineering involve considerable mathematical modeling.

Mathematical theory of the Spread of Epidemics determines the number of susceptible, infected and immune persons at any time by solving systems of differential equations. The control of epidemics subject to cost constraints involves the use of control theory and dynamic programming. We have also to take account of the incubation period, the number of carriers and stochastic phenomena. The probability generating function for the stochastic case satisfies partial differential equations which cannot be solved in the absence of sufficient boundary and initial conditions.

In Drug kinetics, we study the spread of drugs in the various compartments of the human body. In mathematical models for cancer and other diseases, we develop mathematical models for the study of the comparative effects of various treatments.

Solid Biomechanics deals with the stress and strain in muscles and bones, with fractures and injuries in skulls etc. and is very complex because of non symmetrical shapes and the composite structures of these substances. This involves solution of partial differential equations.

In Pollution Control Models, we study how to obtain maximum reduction in pollution levels in air, water or noise with a given expenditure or how to obtain a given reduction in pollution with minimum cost. Interesting non- conventional mathematical programming problems arise here.

Social Sciences

Disciplines such as economics, sociology, psychology, and linguistics all now make extensive use of mathematical models, using the tools of calculus, probability, and game theory, network theory, often mixed with a healthy dose of computing.


In economic theory and econometrics, a great deal of mathematical work is being done all over the world. In econometrics, tools of matrices, probability and statistics are used. A great deal of mathematical thinking goes in the task of national economic planning, and a number of mathematical models for planning have been developed.

The models may be stochastic or deterministic, linear or non-linear, static or dynamic, continuous or discrete, microscopic or macroscopic and all types of algebraic, differential, difference and integral equations arise for the solution of these models. At a later stage more sophisticated models for international economies, for predicting the results of various economic policies and for optimizing the results are developed.
Another important subject for economics is Game theory. The whole economic situation is regarded as a game between consumers, distributors, and producers, each group trying to optimize its profits. The subject tries to develop optimal strategies for each group and the equilibrium values of games.

Actuarial Science, Insurance and Finance

Actuaries use mathematics and statistics to make financial sense of the future. For example, if an organization is embarking on a large project, an actuary may analyze the project, assess the financial risks involved, model the future financial outcomes and advise the organization on the decisions to be made. Much of their work is on pensions, ensuring funds stay solvent long into the future, when current workers have retired. They also work in insurance, setting premiums to match liabilities.

Mathematics is also used in many other areas of finance, from banking and trading on the stock market, to producing economic forecasts and making government policy.

Psychology and Archaeology

Mathematics is even necessary in many of the social sciences, such as psychology and archaeology. Archaeologists use a variety of mathematical and statistical techniques to present the data from archaeological surveys and try to distinguish patterns in their results that shed light on past human behavior. Statistical measures are used during excavation to monitor which pits are most successful and decide on further excavation. Finds are analyzed using statistical and numerical methods to spot patterns in the way the archaeological record changes over time, and geographically within a site and across the country. Archaeologists also use statistics to test the reliability of their interpretations.

Mathematics in Social Networks

Graph theory, text analysis, multidimensional scaling and cluster analysis, and a variety of special models are some mathematical techniques used in analyzing data on a variety of social networks.

Political Science

In Mathematical Political Science, we analyze past election results to see changes in voting patterns and the influence of various factors on voting behavior, on switching of votes among political parties and mathematical models for Conflict Resolution. Here we make use of Game Theory.

Mathematical Linguistics

The concepts of structure and transformation are as important for linguistic as they are for mathematics. Development of machine languages and comparison with natural and artificial language require a high degree of mathematical ability. Information theory, mathematical biology, mathematical psychology etc. are all needed in the study of Linguistics. Mathematics has had a great influence on research in literature. In deciding whether a given poem or essay could have been written by a particular poet or author, we can compare all the characteristics of the given composition with the characteristics of the poet or other works of the author with the help of a computer.

Mathematics in Music

Calculations are the root of all sorts of advancement in different disciplines. The rhythm that we find in all music notes is the result of innumerable permutations and combinations of SAPTSWAR. Music theorists often use mathematics to understand musical structure and communicate new ways of hearing music. This has led to musical applications of set theory, abstract algebra, and number theory. Music scholars have also used mathematics to understand musical scales, and some composers have incorporated the Golden ratio and Fibonacci numbers into their work.
Most of today's music is produced using synthesizers and digital processors to correct pitch or add effects to the sound. These tools are created by audio software engineers who work out ways of manipulating the digital sound, by using a mathematical technique called Fourier analysis. This is part of the area of digital signal processing (DSP) which has many other applications including speech recognition, image enhancement and data compression.

Mathematics in Art

"Mathematics and art are just two different languages that can be used to express the same ideas." It is considered that the universe is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures. The old Goethic Architecture is based on geometry. Even the Egyptian Pyramids, the greatest feat of human architecture and engineering, were based on mathematics. Artists who strive and seek to study nature must therefore first fully understand mathematics. On the other hand, mathematicians have sought to interpret and analyze art though the lens of geometry and rationality. This branch of mathematics studies the nature of geometric objects by allowing them to distort and change. An area that benefits most from the visual approach is topology.

Moreover the study of origami and mathematics can be classified as topology, although some feel that it is more closely aligned with combinatorics, or, more specifically, graph theory. Huzita's axioms are one important contribution to this field of study.

Mathematics in Management

Mathematics in management is a great challenge to imaginative minds. It is not meant for the routine thinkers. Different Mathematical models are being used to discuss management problems of hospitals, public health, pollution, educational planning and administration and similar other problems of social decisions. In order to apply mathematics to management, one must know the mathematical techniques and the conditions under which these techniques are applicable. In addition, one must also understand the situations under which these can be applied. In all the problems of management, the basic problem is the maximization or minimization of some objective function, subject to the constraints in available resources in manpower and materials. Thus OR techniques is the most powerful mathematical tool in the field of Management.

Mathematics in Engineering and Technology

Mathematics has played an important role in the development of mechanical, civil, aeronautical and chemical engineering through its contributions to mechanics of rigid bodies, hydro-dynamics, aero-dynamics, heat transfer, lubrication, turbulence, elasticity, etc.. It has become of great interest to electrical engineers through its applications to information theory, cybernetics, analysis and synthesis of networks, automatic control systems, design of digital computers etc. The new mathematical sciences of magneto-hydrodynamics and plasma dynamics are used for making flow meters, magneto-hydrodynamic generates and for experiments in controlled nuclear fusion.

It is well known that most of the technological processes in industry are described effectively by using mathematical frame work. This frame work is then subsequently used to analyze and comprehend advantages and disadvantages in adopting efficient and novel methodologies in these processes, resulting into the introduction of Mathematical Technology.

The defense sector is an important employer of mathematicians; it needs people who can design, build and operate planes and ships, and work on other advanced technologies. It also needs clear-thinking and analytical strategists.

Mathematics in Computers

An important area of applications of mathematics is in the development of formal mathematical theories related to the development of computer science. Now most applications of Mathematics to science and technology today are via computers. The foundation of computer science is based only on mathematics. It includes, logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory, computer-oriented numerical analysis, Operation Research techniques, modern management techniques like Simulation, Monte Carlo program, Evaluation Research Technique, Critical Path Method, Development of new computer languages, study of Artificial Intelligence, Development of automata theory etc.

All mathematical processes of use in applications are being rapidly converted into computer package algorithms. There are computer packages for solution of linear and non linear equations, inversions of matrices, solution of ordinary and partial differential equations, for linear, non linear and dynamic programming techniques, for combinatorial problems and for graph enumeration and even for symbolic differentiation and integration.

Cryptography is the practice and study of hiding information. In modern times cryptography is considered a branch of both mathematics and computer science and is affiliated closely with information theory, computer security and engineering. Cryptography is used in applications present in technologically advanced societies; examples include the security of ATM cards, computer passwords and electronic commerce, which all depend on cryptography. It is the mathematics behind cryptography that has enabled the e-commerce revolution and information age.

Pattern Recognition is concerned with training computers to recognize pattern in noisy and complex situations. e.g. in recognizing signatures on bank cheques, in remote sensing etc.

In Robotics Vision, computers built in the robots are trained to recognize objects coming in their way through the pattern recognition programs built into them. In manufacturing Robotics, the artificial arms and legs and other organs have to be given the same degree of flexibility of rotation and motion as human arms, legs and organs have. This requires special developments in mechanics.

Computerized Tomography uses the important break through in reconstruction of images of brain and objects from the knowledge of the proportions of photons observed along different lines sent through the object. These proportions can be expressed as line integrals of a function.

Fractals Geometry enable us to design models of irregular objects like clouds, coast lines, lightening turbulence etc. and this uses a combination of probability theory, mathematics and computers. This shows that mathematics can enable us to handle apparently irregular patterns as much as it can enable us to study regular patterns.

In Computer Graphics we find the virtual landscapes and things within them are three-dimensional mathematical objects, and these objects behave and interact according to the equations for the rules of physics that apply within the game. These rules might cover gravity, speed and force, and even stop your character falling through a solid floor but allow them to sink in quicksand. This type of mathematics is used in computer graphics for movies, and mathematics plays an important part in many areas of IT, including programming, designing hardware and project management.

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