Connecting Math to Our Lives Report Mathematics in defence INDIA: Local Project Report from Teacher Mr. G.S.Lawania and his 15 to 16 year old students from the School "Kendriaya Vidyalaya N.T.P.C." in New Delhi.          Index  Mathematics in defence Mathematics education Mathematics in Biology Methematics in medical sciences Traffic flow Informatics and mathematics For mathematics education MATHEMATICS IN DEFENCE Mathematics as a tool for decision maker has always been vital and crucial particularly in the arena of conflict between two opposing forces. With the development of explosives it is possible to inflict harm at a considerable distance as compared to the muscular power of a well built person. This increase in the range of a weapon due to chemical energy compels us to carry out more accurate trajectory computation of a projectile or a rocket. This scenario we find during the first war and to some extent continuing till the end of the Second World War. The study of Ballistics-Internal, External, terminal and Wound Ballistics along with the Hydroballistics had remained a field of active research and development during this period with specific application to warfare. No doubt the research study is still being continued in these areas as there is a requirement to improve these so called conventional weapon systems. The mathematical models of these weapon systems can be well described by the use of ordinary and partial differential equations under prescribed initial and boundary conditions. Thus one may tend to believe that the type of mathematics one needs in defence is confined only to the study of differential equations. This in my opinion is a simplistic approach to the application of Mathematics in Defence. It is important to note that just after the end of the second world war the devastating effect of atomic power in the form of atom bomb was clearly demonstrated to all mankind. At the same time due to research and technological breakthrough in the field of Electronics we have become capable to communicate, identify and destroy the targets thousands of kilometers away from us. With the result today we have entered into the area of Electronic Warfare."'Modern war in the eyes of some people is a push-button destruction. Metaphorically one may say, some of the advanced countries of the world can destroy the enemy within a span of few seconds. Thus it is important to note that in a modern war scenario one has to assess a situation at a much faster rate and at the same time arrive at a decision equally rapidly. Human body and organs are a gift to us from nature. But our organs are not capable to withstand the strain of the rapid flux of events in a modern warlike situation. With the result we are compelled to carry out our research studies and development work to design instruments, which can function like human organs but at a much faster rate. To drive home my point I would like to emphasize the importance of mathematics in the development of robotics, computer vision and pattern recognition. Here the type of mathematics we need is not only confined to the solution of ordinary and partial differential equations under prescribed initial and boundary conditions but also need to know how a three dimensional scene is mapped into two dimensional picture. How much information is sufficient for the computer to Identify the scene from its two dimensional mapping and again reconstruct it if required? The type of Mathematics'" which is relevant in such studies is highly abstract and needs many new concepts based on computational geometry, topology, theory of algorithms and efficient methods of data processing. Today mathematics and computers which play a vital role. in modern warfare go together. It is therefore needless to say that many new branches of mathematics such as Fuzzy Set theory again vital in a modern war-like situation where decisions cannot be taken either way, have grown due to their application in defence. No doubt the mathematical concepts, might have been developed to meet our defence needs, can always be used for peaceful purpose. Mathematics is above peace and war. Mathematics and other subjects Introduction Mathematics today is key to the learning of sciences - physical and biological, technology, social and medical sciences. It is also the language of business, finance, defence, health and planning. Mathematics has permeated society. Computers, new applications and new discoveries have extended the frontiers of knowledge in mathematics. It is appropriate to deal with actual or possible relationships between mathematics and other subjects in schools, colleges, universities etc., especially with the role of mathematics for the learning and teaching of other subjects, and vice-versa including problems of mathematics as service subject. The endeavor is to put forth, the recent developments in mathematics in connection with other subjects or vice-versa. Mathematics education: The problem of mathematics education has been a matter of discussion and debate in every country all over the world. The education must be up-to-date in form and substance, keeping in view the requirements of a growing society. The use of mathematics pervades modern society and its impact, already immense, is growing very fast. It has become an essential component of natural and social sciences. The mathematical formulations of biological problems have led to rapid growth of the subject. History of mathematics tells us that its development is related to the needs of the society. It is this aspect that has to be borne in mind even today. The objective of mathematics education is to impart such knowledge which will not only reflect the beauty of mathematics with all its rigour but will also give an experience of its strength to support the development of other spheres of human knowledge and in solving the pressing problems which a society faces in any given epoch The mathematics today is characterized by axiomatic approach and rigour. Mathematics gives exactness in thinking and provides quantitative approach. Mathematics in abstract form has been accepted as a trainer of thoughts promoting logical thinking and enlarging intellectual horizon of man. This gives mathematics a unique position in education. The general education, for this reason, has always included mathematics and in the present state of development has become indispensable. It is the connecting link between liberal and technical education. Mathematics education must give mental pleasure and develop creativity. The creative intellect is king pin to scientific and industrial progress. The dominant role of mathematics in the modern world is very well reflected in proliferation of mathematics in various branches of knowledge which sometimes ago was not thought of. Mathematics is not confined to academic elite. It is attracting men and women with talent and high intellectual ability. Mathematics in other fields The Mathematical research and teaching at present times has greatly extended. The mathematical techniques were mostly used in mathematical sciences such as physics. They have now penetrated into new fields of technology, into biological sciences, into economics, into social sciences, into atmospheric sciences, into operation research, into ecology, etc. Mathematics in Biology: The subject of mathematical biology given with the use pf mathematics in the study of biological systems Involving physical and chemical' processes, Since these processes have already been studied in physical sciences, it was possible to apply these methods and techniques in other fields of study where these processes are involved. One of them is process of diffusion. It is by this process that oxygen enters the blood stream from the lungs and subsequently to be utilised by tissues for energy systems. The biological processes involving diffusion mechanism can be studied through diffusion equation, which is an equation well studied by physicists and chemists. If C denotes the concentration of oxygen, the diffusion equation is V2C = K a C / at It can be solved for suitable boundary conditions which depends on the blood vessels. The solution in a closed form is not possible. Therefore one has to take recourse to numerical methods and use the computer to arrive at a solution. This would be possible after one builds a model of a particular diffusion phenomenon, study it mathematically and check with the experimentally observed results. Methematics in medical sciences The subject Bio-mathematics in which one studies the flow of blood in veins has made rapid progress through the models describing various types of flow. One makes use of equation of continuity, equations of motion and constitutive equation in the process. These equations have been well studied in fluid mechanics. Diabetes mellitus disease is known, In it sugar concentration in blood rises with the failure of mechanism which regulates the sugar in the blood stream. Differentiat models in respect_of the process are formulated and the mathematical study is made. This study involves the knowledge of elementary differential equations. One studies the quantity of insulin required in a particular case to keep the balance of sugar in blood. Since the sugar metabolism system is homeostatic process, two key parameters can be identified, the sugar and insulin sensitiveness, each to the other. Hence any model which includes these two variables will represent a true situation giving results with a fair degree of accuracy. The mathematical study of tumour, growth, blood pressure diffusion phenomena in bronchial tubes, micro circulatiot etc. has led to conclusions which have helped to find proper treatment. In recent times synovial joint lubrication study has made good progress. It is known that when surfaces are rubbed against each other frictional forces come into play leading to wear and tear of the surface material. This can be reduced by introducing some lubricants between the surfaces. This has been well studied using mathematical models. Differential equations and their applications The study of differential equation can be made exciting by studying its application to 'real world' problems. The applications have two aspects. First the problem to be solved is clearly outlined and one or more differential equations are formed as model. The initial conditions are clearly enumerated and solution obtained. The process can be termed as modelling also. The various types of problems that can be tackled are (1) population models (2) building of three stages rocket (3) flow of liquid from a container (4) laser and drilling holes These are just a few examples. Here the topics in Physics, Chemistry and Engineering have not been included for they are known. The only point to emphasise is that the study of differential equations should include these types of problems which would help in the better understanding of the subject and establish the relationship of mathematics with other subjects. In the process the teacher will have to be well acquainted with the problem in the other fields and develop a model to explain, solve and verify with the observed results. Traffic flow It is interesting to find how equations of continuity and motion in fluid mechanics are applied in the study of traffic flow. One also analyses the traffic light problem and host of problems connected with it. The study of flood waves, study of waves on glaciers, sedimentation of rivers, involves the solution of basic equations of fluid dynamics. Informatics and mathematics Informatics is an engineering science, which is developing its own theoretical foundation. It operate with abstract symbols, developing formal methods and constructing basically mental objects. The main tools of theoretical computer scientists are mathematical ones from algebra, discrete mathematics and mathematical logic, and recently geometry as well as analysis. For mathematics education In view of the increasing application of mathematics in various fields of studies and the necessity for the learners to be exposed with the basic mathematics, it is necessary to design contents of mathematics education in such a way as to meet this need. An outline of mathematics contents to be imparted from primary to university level must be chalked out by mathematicians with the help of scientists' working in other fields and using mathematics. The learning of mathematics must incorporate in it the application aspect in a proper way which should generate in the learner's enthusiasm, curiosity and excitement in solving challenging problems of everyday life. To achieve this objective it is necessary to develop a comprehensive educational discipline of applied mathematics, attracting brilliant young students with a prospect of promising career. Applied mathematician should be capable of promoting interaction. I am of the view that education of applied mathematics should be developed separately to meet the challenge of problems, in other branches of knowledge, to be solved by use of mathematics. It is suggested that an outline of the basic program of mathematics education should be drafted. There should be continuous interaction between users of mathematics and professional mathematicians. Mathematics course must incorporate in it, the application aspects in good measure. Emphasis must be placed on the solution of real world problems. The rigorous mathematical formalism may be put aside to solve these problems and it can be taken care of by pure mathematicians interested in application. The courses should be designed for users of mathematics e.g. course in mathematics for biologists or economists. It is not an easy task but experts must pool their experiences to evolve a course of study in applied mathematics which could meet the requirements to a great extent. The making of such a syllabus is not easy and will vary from one place to the other, depending on the facilities available, interest of the mathematicians and field of interaction available. But the subject and its contents could be developed on its own and researches carried out on important problems. This conference can accomplish this task by evolving the contents of mathematics education at all levels to meet the challenge of present time.