STUDENT'S CORNER

Geometry in day-to-day life...

Study of geometry provides basic fundamental skills and helps to build logical thinking, deductive reasoning, analytical reasoning and contributing to holistic development.

The best use of geometry in daily life is in the construction of dams, buildings and bridges.
The audio-visual presentation in different fields such as education, entertainment, etc. uses geometry as a creative concept.

This concept is used by astronomers to track stars, measure the orbits and distance between planets and satellites.

Geometry

Geometry in ancient Greek (geo - meaning earth and matron meaning measurement) is a branch of Mathematics primarily concerned with the shapes and sizes of individual objects, their relative position and spatial relationships. How has it evolved...? The existence of geometry can be witnessed from the era of early men. At that time this subject doesn't have any existence, but the use of geometrical concepts was traced from the fossils, ruins and artefacts. The ancient people devised mathematical rules and techniques useful for surveying land areas, constructing buildings and measuring storage containers; extending the practical knowledge and formulated the abstract subject now known as "GEOMETRY". Types of Geometry:

1. Euclidean geometry

Euclidean geometry is used to understand the fundamentals of geometry and its relationships between length, areas and volumes of physical objects. It refers to the study of plane and solid figures on basis of theorem.

2. Non-Euclidean Geometry

Non-Euclidean geometry differs in its postulates on the nature of the parallel lines and the angles in the planar space as validated by the Euclidean geometry.

(a). Spherical Geometry - It is the study of plane geometry on a sphere. Lines are defined as the shortest distance between the two points that lie along with them. This line on a sphere is an arc and is called the great circle.

(b). Hyperbolic geometry - The Non-Euclidean geometry of Gauss is usually called the Hyperbolic Geometry because of its very natural analytical models. It is the geometry of space in which Euclidean's theorem fails.

Mathematics And World

It is well said that a single step can change one's life, but most often, people are confused and unable to take up that step. If somebody says that solving a mathematical problem, can help you take up that step then sorry to say but it might not work. After a long analysis it has been found that that particular step is about generating a better understanding of life than waiting for something magical and magnificent to occur and so is mathematics. It is not just about numbers, equations, computations or algorithms but about understanding! understanding everything, from bill of groceries to making spacecrafts, from a way to smile to a way for happy life. In short if we define mathematics and life then we can say life and mathematics are equivalent.

When it comes to measuring importance of anything, we often go by measuring its applications, usefulness in daily life. There are numberless applications of mathematics in field of Science and Technology like design and analysis of control systems for aircraft, computer programming, cryptography, understanding motions, radio interferometry, aerodynamics, terrain modelling, cryptology, analysis of statistical data, speech recognition, computer design networking, estimating of ocean currents, electromagnetic analysis for detection by radar and many others.

But usually, people are not convinced to respect mathematics for its significant applications they still consider it theoretical which has to do nothing with our lives.

But people need to understand that learning algebraic skills help them to earn more money in their daily lives, geometry and trigonometry can upswing their sports performance, equations can help us view real life problems in an easier way. These were some of the places in daily life, where, if mathematics is applied then things turn pretty easy to deal.

Above mentioned applications were just the direct implication of mathematics but other than these mathematics helps us enhance our analytical and problem-solving skills, creates the basis for systemic thinking, improves the skills required to arrive at a logical conclusion, expands the mind to handle unfamiliar tasks with ease and confidence. So, it won’t be wrong to say that Mathematics is a tool via which we get a better and easy life.

If you still question the existence of mathematics, then stop doing it because real life is pure mathematics and the moment you will start realizing the depth of statement you will stop questioning existence of Mathematics.

Career options in Mathematics

The scope of Mathematics is still largely untapped in India as most math lovers are still unaware of its prospects and how it can be pursued. Most students who are looking for a career in Mathematics can go for different choices. Across various industries, a career in Mathematics in India opens up plenty of opportunities. Today, a person with exceptional Mathematical skills is highly desired in the market.

Digital Signature

Digital signature is a mathematical theme used for the validation of the legitimacy and integrity of a message, code or digital documents which works on the principle of mathematical algorithm. It is the digital version of a written signature but secured one which helps in identification of origin, identity and tracking electronic documents, transactions or digital messages. It's merely associate encrypted message that exclusively the receiver can decrypt.

Working of digital signature

It works on the principle of public key cryptography, conjointly referred to as unsymmetrical cryptography. By using public key formula, like RSA (Rivest-Shamir-Adleman), two mathematical connected combination of keys are generated, one personal and one public key combinations are created. The individual United Nations agency creates the digital signature uses a personal key to code signature-related information, whereas the receiver uses the signer's public key to decode that information.

If the recipient cannot open the document with the signer's public key, that indicates there is a drawback with the document or the signature, this is often however the authenticity of digital signature are detected.

Advantages of digital signatures

• It is worldwide accepted and legally compliant, the general public key infrastructure (PKI) commonplace ensures the creation of vendor-generated keys and keep firmly attributable to its acceptance internationally and because of which, a large number of countries are accepting it as legally binding.

• Digital signatures save tons of your time from long processes of physical document signing, storage and exchange and also permits businesses to urge fast access to sign documents.

• Organizations and establishments will go paperless and economize that were spent on the physical resources and on the time personnel and workplace area.

• Reducing paper use conjointly cut back the physical waste generated by paper and also the negative environmental impact of transporting paper documents

• Digital signatures create an audit path that helps internal record-keeping easier for businesses. With everything recorded and keep digitally, there are fewer chances for a manual signee or functionary to form miscalculation or misplace one thing.

How to get digital signature?

To create a digital signature, signing software, like email program, are needed to offer a unidirectional hash of the electronic information to be signed. A hash is a fixed-length string of letters and numbers generated by an algorithmic formula. The digital signature creator's personal key is then used to decrypt the hash. The encrypted hash - alongside different data, like the hashing formula -- is that the digital signature.

The reason for encrypting the hash rather than the whole message or document may be a hash perform will convert an arbitrary input into a fixed-length price, that is sometimes at short. This protects time as hashing is far quicker than sign language. The value of a hash is exclusive to hashed information. Any modification within the information, even a modification in a very single character, can lead to a distinct price. This attribute permits others to use the signer's public key to decode the hash, to validate the integrity of the info. If the decrypted hash matches a second computed hash of a similar information, it proves that the info hasn't modified since it had been signed. If the two hashes do not match, the info has either been tampered with in a way and is compromised or the signature was created with a personal key that does not correspond to the general public key bestowed by the signer a difficulty with authentication.

Disadvantages of Digital Signatures

A digital signature is mostly dependent on the technologies used to create it. If the technology advances it works faster, then digital signatures are needed to change quickly or lose their functionality.

• For using digital signatures, you will be needed to purchase digital certificates that can be quite costly.

• Users are also needed to purchase verification software.

Mathematics - An Insight For Student

"Mathematics is the gate and key of sciences. Neglect of mathematics works injury to all knowledge since he who is ignorant of it cannot know the other sciences or the things of the world."

The above words of Roger Bacon, an English scholar aptly depict the need and contribution of mathematics in the world. In the world of millennials where everything is controlled by a single click, students struggle to digest the flavor of mathematics presented in the classrooms. Their shining faces and sparkling eyes always have the curiosity to know further but with confusion regarding the significance of content. The kind of mathematics, we study as today's curriculum is the compilation of work of many prominent mathematics over the centuries. None of them developed these results in the way we used to study them.

Follow your intuition, work with dedication and discuss the problems. It will not only help you in a better understanding of the subject but also nurture you with an analytic mind, clarity of thoughts, problem solving ability, and effective communication skills. The branch of mathematics like computational science, applied analysis, optimization, and differential equations, etc. play a vital role in the industrial field. Mathematical and statistical modelling are essential in engineering, physical sciences and contribute significantly to biological science, medicine, economics and commerce. The mathematical tools such as discrete mathematics, number theory, cryptography, etc. are the foundations for creating a secure internet environment. In reality, from cradle to grave life is beautified by mathematics. Even Galileo defined Mathematics as a language in which God has written the world. Mathematics is not a one-take wonder but requires lots of patience, practice and honest efforts. As Archimedes said, "Mathematics reveal its secrets only to those who approach it with pure love for its own beauty". So, if you dwell in the ocean of mathematics, you will surely get the pearl of wisdom. discovery, and invention.

It is rightly said, "The best way to learn mathematics is to do mathematics."

Mathematical modelling is an art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application. Mathematics is the mother of all sciences. It has its branches widespread in almost every field you can name. Similarly, mathematical modeling has widespread usage in every aspect of our lives. It is such an indispensable part as it is an important step from a theoretical mathematical training to an application-oriented mathematical expertise. It is simply a language related to problems, numbers and for more. Applications of mathematical modelling gives precision and direction for problem solution. ranging from Anthropology for modelling, clarifying and reconstructing skills to Architecture in creating virtual reality, from Arts for computer animation to Astronomy in evolution of stars, detection of planetary systems, origin of universe, from Chemistry for molecular modelling, electronic structure calculations to Criminalistic Science fingerprint or face recognition, from Economics for labour data analysis to Electrical Engineering in stability of electric circuits. The above mentioned are a few of them. The other fields include Finance Fluid Mechanics, Geosciences. Internet Linguistics, Material Sciences, Medicine, Meteorology, Muric. Neuroscience, Pharmacology, Psychology. Space sciences etc. According to Einstein - "A good theory should be as simple as possible not simpler."

If there is no MATHEMATICAL MODELLING. what is the point of evolution and existing of universe?

Mathematical Modelling: The Way of Living

Game Theory

Mathematics is not always about equations, numbers, algorithms etc. It is about understanding and the art of giving same name to different things. Numerous mathematicians have proved the versatility of mathematics. Game theory is one such amazing and unique application. It is a branch of applied mathematics most commonly used in Economics. However, it can be very successfully applied to other social sciences as well as Evolutionary Biology. It gives both descriptive answers (what people do) and prescriptive answers (what people should do) in a given game. Question is why is it relevant?

Game Theory is a very good tool in predicting outcomes, not only in the very simple games, but also in predicting the outcomes of evolutionary strategies and of predicting outcomes for signaling games which can inform us human and animal communicative strategies. Running iterated games over populations can introduce interesting qualifications to these very simple ideas as well.

Primarily, A game needs 3 things before game theory can be applied:

1. Players

2. Strategies

3. Playoffs

Game Theory

Here is a simple example of a coordination game: The Driving Game

Left Right

Left 100, 100 0, 0

Right 0, 0 100, 100

n (no. of players) = 2

Imagine two cars driving along a road towards each other. Each driver has the choice of whether to drive on the right or the left. The table represents all possible outcomes. If both players decide to drive on the left (or right) then they will pass each other and each get a payoff of 100, however if the players decide to drive with one on the right and the other on the left then they will collide and both players get a payoff of 0. What should the players do? In the situation of a one-off game where the players are not aware of the others decision the choice is 50/50. In an iterated game, however, the tendency will be to choose either left or right and stick with it, this will allow the other driver to co-ordinate so both players get the maximum payoff. Once both players are either playing left every time or right every time neither will switch because both know that switching would probably result in quite a nasty crash. These strategies, where nobody has anything to gain from switching, are known as NASH Equilibrium.

Ever think life without traffic lights?? How much quos on roads it will be or how problematic it sounds!! Roads without Traffic lights mean Pizza without topping. Sometimes, traffic lights didn't work; so many accidents took place or traffic jam held. So, how these traffic lights actually work! Ever Think??

Nowadays, economic and social development is highly dependent on good transport system. Indeed, more people and products need to travel longer distances, the traffic flow is larger as well as the risk of accidents, congestions and their negative consequences such as loss of human lives, traumas, etc.

That is why safe and efficient transport system is needed. The safety at the congested crossings, traffic lights are very important because they decide which lane has a priority and for how long. However, the duration of this priority shall be defined for guaranteeing the efficiency of the system. So, optimization models are an important tool for proposing a solution to this decision problem. There is a direct connection between traffic lights and mathematicians. How? Let's see!!

Traffic lights could be made a lot smarter by applying mathematics, as researcher Rene Haijima discovered. His model shortens waiting times and thus prevents frustrations. The roads are empty, you are almost home. Better yet: you could have been home a long time ago, if all lights had been green.

Mathematics in Traffic Lights

Traffic modeling has been of interest to mathematicians since 1950s. Research in the area has only grown as road traffic control presents an ever-increasing problem. Generally, models for traffic flow in road networks are time-dependent and continuous, that is, they describe traffic by a continuum rather than as individual drivers or cars. These macroscopic models describe the temporal and Mathematics in Traffic Lights spatial evolution of traffic density without predicting traffic patterns of individuals.

Most existing continuous models consider unidirectional traffic; thus, the traffic density depends only on a single spatial dimension. The governing equations in this class of macroscopic models are inspired by gas dynamics equations. “Traffic lights are a necessary tool to redirect the traffic flow within road networks and therefore offer the potential to mitigate congestion even for high traffic volumes based on mathematical insights.

The simplest analysis used for timing a traffic signal is the calculation of the traffic volume to capacity ratio.

Let's understand this through an example of an intersection of two one lane, one-way streets. The basic theoretical capacity of that intersection is 1800 vehicles per hour. Each lane then would have a theoretical capacity of1800 for an hour of green time.

For planning purposes, a realistic capacity of1400 vehicles per hour is often assumed. Therefore, if each roadway had 700 vehicles approaching during an hour and the approach speeds on each roadway were the same, then the green time would be divided equally between the two approaches. The yellow change interval timing is based on the speed of the traffic on each approach. Typically, the yellow change interval is set at 1 second per 10 miles per hour (1.6 km/hr) with a minimum of 3 seconds.

So, if the vehicle approaches to the signal have 30 mile per hour speed limits, the yellow would be set at 3 seconds. The yellow interval should then be followed by an all-red clearance interval to allow the last vehicle to clear the intersection before the approach gets a green.

For crossing a 1 lane roadway, all red can be very short. For most intersections, particularly those on high-speed roadways, vehicles detectors are used to control the length of the greens and to determine the proper time to end the green. Yellow change intervals could be as high as 6 or 7 seconds and for a large intersection with a wide median all red clearance could be 3 or 4 seconds. For more complex intersections there are many more variables to consider.

So, basically mathematics is everywhere. Mathematics may be pure, but it is applicable in day-to-day life, it's just we need to see how and where mathematics can be used.

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