ROBOTS

Grassroots Robotics

560 Indiana teachers receive VEX robots & training Thousands of additional students learning robotics & competing.

A new and rapidly expanding robotics program in Indiana is enabling many thousands of high school students to design and build robots—and compete in robot challenges—as part of their school curriculums. This is an exciting development because the students get to “wrench” on robot designs of their own creation even as they learn the diverse disciplines that come together in robotics—mechanical engineering, electricity, math, materials science, physics and more. This rapid expansion of robotics in Indiana schools is the result of carefully planned robotics workshops and competitions held by the Indiana Robotics Educators (IRE) in various Indiana locations throughout the year. 

Left to right: Dan Ward, Luke Ward and Kyle Wiley of the IRE are immersed in robotics. Two VEX robots are visible in the foreground, and in the background is a retired industrial robot that was formerly used by Chrysler to move transmission casings.

With financial support from the Indiana Department of Workforce Development (IDWD)—a state agency—IRE is able to provide free workshops and competitions that benefit teachers, students and schools. At the end of the three-day teacher workshops, the teachers get free software, curriculum guides and a VEX Starter Kit each to take back for use in their classrooms.

As part of the deal, teachers must field one or more student teams to compete in robot competitions held locally in the state. This has been a boon for the teachers because the students are extremely enthusiastic and highly motivated, and the resources for the new robotics programs are almost entirely paid for by the state, i.e., by taxpayer dollars. The IRE program costs teachers only two or three days of their time at a training workshop; the teachers and their school boards are not asked to pay for anything.

Backing for the IRE’s robotics programs has come from the IDWD because that agency is charged with ensuring a flow of technically trained graduates into the local workforce. IDWD partners in this program with Ivy Tech Community College of Indiana—a statewide college system that provides the facilities for the teacher workshops and that ultimately trains technical majors for Indiana industry.

Everybody Wins

This expanding program is a win-win for all. It has long been recognized that robots are a learning accelerant—a catalyst that galvanizes students’ interest in STEM subjects (science, technology, engineering and math). Because robots are entertaining and fun to build, operate and compete with, students progress more quickly in their studies and become more aware of technical career opportunities along the way. From the viewpoint of engineers, educators and government, this is of critical importance in a changing world in which our technological leadership is being challenged. 

Teachers progress in their professional development and the performance of their schools is recognizably enhanced. The Ivy Tech campuses where the workshops are held benefit through a strengthened working relationship with the teachers who are educating the students who will soon be going to college. Even hobbyists in general benefit because the program heightens the demand for robotics technologies that squarely overlap with those enjoyed by hobbyists of all ages; in other words, the robotics market itself is expanding as a result of the IRE initiatives.

The Leaders behind the IRE

Dan Ward is the Design Technology Program chairman at Ivy Tech, the only community college system in Indiana. Dan and his brother, Luke, an assistant professor at Ivy Tech, have been pioneering IRE robotics educational programs since 2000 with the goal of inspiring high school students to pursue technical careers. Kyle Wiley, assistant professor at Ivy Tech, joined their team, and what they have achieved in two short years is nothing short of phenomenal. Their accomplishments could serve as a role model to change the landscape in technical education nationwide to our country’s benefit. In fact, Dan and his IRE associates have been contacted by educators in several states who wish to replicate the successes spearheaded there in Indiana.

Robot: When did you first launch a robotics program in Indiana, and how did that come about?
Dan: I was on a visit to the Indiana Statehouse to give a robotics demonstration to our legislature. I was there with my brother, Luke, who works with me on grant projects. We were trying to convince the state legislature that it was a good idea to fund educational robotics. To set the context, the Indiana Workforce Development agency has been a big supporter of FIRST in our state. We were there with our FIRST machines, probably 10 teams. We were offering quite a show, making a lot of noise; a lot of robots were moving around. 
     The people from Workforce Development had heard we were coming and stopped by and asked what was the biggest hurdle in getting more schools involved with robotics programs like FIRST. I said, “training.” That’s the first hurdle; the second is money. They asked, “What is the best way to solve this?” And I told them “Let us train them. We will use your funding and give the teachers the equipment they need so that they won’t have to get it from their school.” 
     They then asked how much it would cost; now, I was on the spot. I turned to my brother and we whispered back and forth and then I turned around and said that it would cost in the vicinity of X dollars. Based on that, they figured that their cost per teacher would only be around $300 for three days of training and all of this equipment. They had never seen anything like it and asked me to commit this to writing in a proposal. This was on a Tuesday, and I said I could have it to them by Friday, so I spent a couple of sleepless nights putting the proposal together.

Robot: What happened next?
Dan: The following week was business as usual until Wednesday, when we jumped in the car to drive to Atlanta for that year’s FIRST national competition. While we were there, my cell phone rang; it was our school business office manager at Ivy Tech, who said that a check had shown up with my name on it and that it was for a substantial sum of money, and nobody knew what it was for. The DWD thought the proposal was such a good idea they jumped on it in less than a week! When that later played out, we actually surpassed our first year’s projected numbers; we came in under budget.

Robot: When and where are the workshops held?
Dan: Teacher workshops are held in the summer, June through August, on Ivy Tech campuses around the state. We have a 10-week window to hold eight workshops.

Robot: Can you characterize the process of writing a grant? What do you put into it?
Dan: Those in a position to allocate money want to see sustainability; that’s first. The grant writer must say that with the history here, there is this much more we can do, and that it will be self-perpetuating—that we have people who are involved and that they are going to stay involved. That’s the important part. Even if you don’t have it at the moment, you have to prove that it will be sustainable in the end.

Robot: How many have you trained, and when did your workshops begin?
Dan: The first year, we trained 35. That was a proof of concept on an old grant. We did that on donations; I put a little money into it. We scraped and got the information together. We had really good luck. The teachers really seemed to enjoy it. 
     The next time around, summer of 2006—and that is the same time as we partnered with the Workforce Development people—we had 121 participants. That was without advertising—just word of mouth. We had requirements defining what we could use the money for. We were able to get 121 people representing 102 high schools. 
     In 2007, we did a little marketing and sent out some emails. The Department of Education had a mailing list, and the science technology teachers received this on their list serve as well. I think word of mouth did more for us than any other channel, quite honestly. At this point, we have trained 560 teachers—560 robots have been delivered into teachers’ hands.

Robot: What exactly do teachers get in these workshops?
Dan: All have been given free VEX Starters, current Autodesk Inventor software, a DVD with a curriculum that we have developed over the years and robot project and competition notes. This is provided in Microsoft Word format so that they can edit it. They also build a robot during the workshop and compete with it on the third day.

The Tickler an Involution Champ

The Tickler, assembled from a single VEX Starter Kit, is a winning design that has performed well playing Involution.

Sometimes, a design functions so well that it lives on season after season to continue competing in the game it was designed for. The Tickler, designed to play a game called Involution, can be assembled using just the pieces in a single Vex Starter Kit.

When playing Involution, robots collect racquet balls and put them into a goal made up of several pipes of differing sizes. The lowest pipe stands 2 inches while the tallest is 10 inches. The Tickler was designed to use motors effectively by making them perform multiple tasks. One motor collects balls and lifts them up 10 inches. This motor's function is the key to this design.

The unique feature of the Tickler is the way it is able to collect and score. Collection is achieved by means of a "beater brush" mechanism that works like a farming combine. The robot drives around the playing field and gobbles up the balls. Brushes made from the zip-ties included in the kit spin and draw the balls up into the robot. The balls then proceed to two more brushes that pass the balls between them until they are delivered to a storage ramp on top. As they are initially drawn into the robot, the collected balls are funneled into two columns instead of one to avoid jamming the mechanism. These two columns continue to the top of the storage ramp. To score, the robot positions itself above the desired pipe and lets one column at a time discharge a ball by opening a servo-operated gate. This robot can collect, store and score 16 balls at a time.

Robot: How to you administer these programs? 
Dan: I have partners around the state, and this is a big reason for our success. Our Ivy Tech Community College is the largest school in Indiana by enrollment, and we have 24 locations. I’ve been working at Ivy Tech for 12 years and know pretty much everybody, we are all friends. We work together on curriculum issues and meet a couple of times a year, sometimes on holidays—all sorts of things. So I call my counterpart and say, “Mike, we are coming down to your campus for the robotics competition, and I need you to help get us set up.” He then goes into motion, rents us a motel, arranges for food and coordinates with the instructors who are coming in. 
     When I’m taking a workshop to a campus, I have to work closely with the chancellor of that region as we ask them to fund a small part of the activity. I note that in addition to letting us come in and use a building, I would like my counterpart to be paid for the week, so that he or she can focus and help me because I need that person to be available in case I need something. I’m asking them to invest. When they see 30+ teachers in there, working, at their campus, having a good time developing their skills as educators, they can appreciate the investment that they made to the community.
     While the programs are ongoing, we are audited internally and externally, which is fine because we are spending taxpayer money. I also have to write grant reports that specify the outcomes and results of the workshops.

Robot: How do people sign up for the workshops? Where do you publish calendars? 
Dan: We announce our competitions and our workshops at .robotevents.com, which is where participants can sign up as well. I can’t say enough about Robotevents.com because I truly think it is going to be gigantic. We are going to have people register through this engine. Registering by email is the worst thing that one can do; getting this done through a Web portal is more efficient as you are not flooded with questions from people who are not going to commit. 
     I put the information out there, and they can contact me if they have a specific question and they can register directly through the website. I can manipulate the reports from that website and manage who is going where. I can generate reports without having to do any typing. Robotevents.com is a great partnership between the sponsoring companies: Autodesk and Innovation First. I think you will see more people like me, who will start using it in that manner—to announce their conferences and workshops. It will be a giant clearinghouse and really do great things for people who are interested in competitions and getting in touch with other people.

Robot: How is the upcoming season shaping up, and how far along are you, overall, in terms of getting a robotics curriculum into Indiana high schools?
Dan: This season is shaping up well. We are organizing workshops in eight strategic locations to get as much done as we can in the time available. Student competitions are growing rapidly; as far as I know, we hold more than any other inland state. The Indiana Robotics Educators will be providing a number of challenging games for teams to compete in, including Involution and Marbelous. But as fast as we’ve grown, we have hit only about half the schools that we will ultimately connect with in Indiana. And the good news is that educators in other states are becoming aware of the IRE programs and are seeking guidance to create similar programs.
     At a time when competence in engineering and hard science is defining the future success not just of students but also of businesses and even nations worldwide, Dan’s robotics in education initiatives are significant. Without even thinking about it, students dive into the technical areas to become better robotics competitors, and many have gravitated to technical majors as a result. We take our hats off to the achievements of Dan and his IRE associates.

* Below, in an extended interview, Dan further expands on the philosophy and methods that have made him so successful in expanding robotics education in Indiana. His insights should prove invaluable to others striving to expand robotics curriculum in middle and high schools anywhere in the nation.

--Tom Atwood

Workshops

Robot: Can you tell us a little more about the package that you provide to teachers in your workshops?
Dan: What is exciting is that we have integrated Innovation First's starter VEX kit into our curriculum. We talk specifically about VEX systems. We do talk about other robotics programs and systems but we are primarily focused on VEX because it is so affordable and successful in this context. We give teachers information on lots of different robotics competitions and different things that the teachers can do in this arena as well.

Robot: Do you bring any non-VEX robotics technology to the workshops?
Dan: We show teachers a variety of robots including Parallax BOE-Bots, a FIRST robot, a BEST robot, VEX systems and a LEGO Mindstorms, among others. We put them at the front of the room and leave them there for the duration of the workshop. Size does not matter here; they all work basically the same way. These take up a lot of room but it makes an impact on the teachers. When you let the teachers drive these robots it impresses them; they remember that forever. 
     The content of the workshops would take hours to describe. The agenda seems straightforward but there is a lot there and it is digestible in the three days. We figured it out--all it took was a little trial and error and a lot of thought. And we constantly modify what we talk about and the materials to make it more friendly and easier to understand; we are still learning. We see the results from it; we see the people competing and they keep in contact with us and ask us what’s new and what can they do next? 
     My goal is to try to help people who truly want to help others.

Robot: Do you include middle schools in your grant programs as well as high schools?
Dan: When I write grants I make sure that we can include middle schools. By the time you get to high school it is too late. Kids learn so fast when they are in third through sixth grades. That’s really where it should first happen .

Year 2 Workshops

Robot: You mentioned that teachers may attend workshops a second time the following year; can you tell us about that?
Dan: There is an interesting aspect of our program with respect to sustainability. It is not just new teachers who attend our workshops—veterans also come back for another workshop. In a sense they are our competitors. They are going to buy equipment for their school, and they are going to approach their school board and say that there is a need to further develop robotics and competitions at their school.
     The second year programs are distinctly different. In our first year of grant-funded workshops in Indiana (2006) we had 121 people. Last year (2007), we had 156 new people and on top of that, we had 35 – 36 veterans who came back for a second year. That’s not to say that if you don’t come back a second year you are not going to further the cause. Some people just want to take it a step further, and that’s fine.
      In the first year workshop, teachers learn the basics, including how to build a robot and compete. In the second year, teachers learn a little bit of programming. They focus on sensors, and on making those sensors and programming work together to make the machine do a task. They come initially to the same location as first-year people and they spend a few hours together. The veterans talk about what they have done over the past year with their robots, and then they disappear to a different area.
     This year we will incorporate a work cell programming project in the second year program. Teachers will learn how sensors work and how to integrate sensors offered with the VEX robot into a work cell that we have designed. After two days, we bring them back into the rookie workshop and let them demonstrate what they have built. We split the rookies into teams of three or four, and we have the veterans mentor the rookies, who are building robots for the competition at the end of that day.

VEX Robotics in Indiana Schools

Robot: What percentage of the high schools in Indiana are using VEX robotics?
Dan: The goal of the grant is to have a VEX machine, a robotics presence, in every high school in the state of Indiana. We think we are approaching half of this. 
     Moving forward, this year we are direct mailing to the schools that we haven’t heard from. We are emailing superintendents and principals, and if we can find online the teachers who are involved in science, technology and mathematics, etc., we are going to email and/or direct mail to them at their school. We have already started this process.

Robot: How many VEX robots ought there to be in any given school?
Dan: I think you can thoroughly engage a class of 22 – 23 kids if you have four machines. With any of these robotics competitions, it’s not just the “build.” There is the design, the build, and then there are the many practical issues of getting ready for a competition. There are lots of things to do, and you can easily engage three or four kids per machine, basically.

Robot: Do teachers who take these workshops purchase additional machines?
Dan: Yes, they will go to their school board and say they need to integrate this into their classrooms. In all honesty, that takes a lot of bravery, anymore. Yet, we’ve found that the schools have been pretty supportive.

Robot: What about school administrators?
Dan: The school administrations take note because they see something that people are excited about. They see that the teachers are very excited and have engaged a large group of students in a short period—it must be worthwhile and it's something they need to take a closer look at.

Robot: Have parents shown interest in robotics programs?
Dan: We’ve found that when students get involved, parents step up, also, and show an interest in robotics. At competitions, parents often say, “I’ve got to get one of these for my kids.”

Staffing and Coordinating Workshops

Robot: What kind of staff is required to hold a workshop?
Dan: My brother, Luke, is also a teacher. He and I handle the workshops with another partner, Kyle Wiley. The three of us basically link arms and work together to get all the material ready. I also have partners around the state and this is a big reason for our success. Our Ivy Tech Community College is the largest school in Indiana by enrollment, and we have a serious advantage—we have 24 locations.
     So, if I want to put on a workshop, say, in Evansville, which is almost a five-hour car ride, down in the southwest end of the state, I get on the phone and call my counterpart at that campus. I’ve been working at Ivy Tech for 12 years and know everybody, we are all friends. We work together on curriculum issues and meet a couple of times a year, and get together for Christmas—all sorts of things. So, I call my counterpart and say, “Mike, we are coming down to your campus for the robotics competition and I need you to help get us set up.” He then goes into motion; he rents us a motel, arranges for food and coordinates with the instructors who are coming in. He takes care of room arrangements, gets us keys—he does whatever is needed to get us set up on his campus.
     Why does he do this? I’m coming down to his campus and I’m bringing in 25 to 35 high school teachers from his area, to his building. The equation is: we tell the teachers that you come to the Community College Campus, our instructors are going to show you something that you have never seen before, potentially, and you are going to learn and be given this equipment to take back to your students. It’s not going to cost you anything but gasoline.
     The teachers are now in love with my counterpart down there, they are in love with the campus and they love the school. Hopefully they will have a favorable experience; that helps us, and it’s a marketing issue, too, for us.
     But, a lot of schools [nationally] don’t have that luxury. When they want to do training, they’ll say, hey come to our college, we are talking a 3-day robotics workshop, here, in this one spot. “Come to us.” But, you know, even if it’s free, they have to pay for their travel and lodging, for food, and it ends up costing them $600 - $700. 
     In our case, all they have to do is come to me, and I’m going to come to places that are within a half hour drive of their houses. That’s a major advantage—I don’t have to worry about facilities. We have laboratory space and can easily set up our computers; we have tables where we can work, power, places to plug in. All of this is duplicated at our campuses all around the state. 

Selling Colleges on Holding Workshops

Robot: How do you persuade the colleges to allocate the resources for the workshops?
Dan: When I’m going to a campus, I explain to the chancellor of that region that we are coming and that I would like him or her to pay for this, this and this. In addition to letting us come in and use your building, I note that I would like my counterpart to be paid for the week, so that he or she can focus and help me, because I need that person to be available in case I need something. I’m asking them to invest. When they see 30 teachers in there, working, at their campus, having a good time, they forget all about the couple of thousand dollars that they had to put forward, if it’s even that much.

Robot: Do the colleges directly benefit from these workshops?
Dan: Yes, because the high school teachers are on campus, this ultimately furthers the flow of their students to the colleges.

Importance of Grant Reports

Robot: What role do grant reports play in this process?
Dan: The workshops are a good investment for the colleges, and part of the payback is that I mention them in the grant reports that go back to the state and to all involved. I say listen, their campus showed us a good time. They did a great job, their campus is beautiful, they are very supportive of this and we appreciate that, because they are helping us spread the word and advance technical education. The grant reports let everybody know. 
     It’s required that I turn a report in on the grant. The agency wants to know where we are at. They want our numbers, who was there and how many versus last year—they want me to compare and contrast just a little bit, but I don’t have to go into too much detail. These people watch us; we get audited two or three times during the course of a season. And that’s fine; they are obligated to do this because it is taxpayer money. They follow our progress and it has become an automatic procedure.

Robotevents.com

Robot: How have you used Robotevents.com in the course of developing workshops?
Dan: If you look at www.robotevents.com, you will see the Indiana Robotics Educators logo. I work with both Innovation First and Autodesk, and both are major sponsors of this web portal. I am in a unique position. When Autodesk and Innovation First first began talking about this website, they wanted me to be a part of it. They wanted my advice and help to prove that this portal would work, and it certainly has worked well for me.

Changing Work Environment

Robot: Are there any other aspects of your grant work that you'd like to share with our readers?
Dan: A couple of things are happening. Indiana is kind of a unique case. We have a lot of automotive industry here, and the fuel industry up north, and we had a lot of heavy manufacturing going on in this state in years past. Since NAFTA, it’s gone. Since the early 1990s, tens of thousands of jobs have gone. Generations of families that worked at these places, and they are now closed. 
The government seems to have embraced that it’s ok if the U.S. turns into a service economy. That’s a lower class citizenship than we want to be at, I think. People want to be successful—I have four kids. I’m looking down the road; what are my kids going to be doing when they are my age? Did I have it better than my parents? Yes, a little bit. My father was first generation college and became an engineer. My kids will go to college but will they be able to make enough money to have a better life than I have given them?
     Today, many companies are appealing to the government for more visas for more overseas workers, because we can’t find enough here. I read Tom Atwood's PowerPoint presentation at Botmag.com on “Growing an Industry,” and thought it was dead on [presentation to Carnegie Mellon robotics educators conference on the state of the robotics industry in the U.S. and where our educational system stands in meeting workforce needs].
     I applaud the working class and the skill it takes to build things. I told my son that when he’s old enough, I’m going to make sure he learns how to weld. At my high school, where my kids attend now, we used to have a beautiful shop. You could take classes in welding, woodshop, automotive, etc. and I thoroughly enjoyed it.
     Robotics is a way to bring the "applied arts" back into play. Look at it this way: robotics teaches you about materials science, electricity, chemistry, mathematics, statics and strength of materials—what other discipline teaches you about all of these things? And the students are having so much fun they don’t even know they are learning in the process.

Importance of Competition

Robot: How important are robot competitions?
Dan: We focus on competitions, because that is where it ends up. After three days at one of our workshops, we have the teachers play a game. That’s what students want to do as well—they want to compete. Without that element we would not have as many students involved. There needs to be an “Aha” moment at the end, some glory that some will achieve. In my 2006 Grant Report you will see the list of program outcomes that teachers are required to give us. They must produce a lesson plan; they have to do 15 things for us to satisfy the state government, to show that they have done what we told the state government they would do. One of those things is to come and compete with the robot.
     Here’s what we do: they can’t use more than what is in that box to play the game that we are going to play. We call it a “one kit competition.” When we realized we could get 100 people to show up every time we did one of these, we were thrilled and nicknamed this the “one kit wonder.” This keeps it affordable for the teachers, who can come and compete and not have to spend any more money.

Robot: It is interesting that you will build robots from scratch at the workshops.
Dan: The teams of teachers have to form an identity. These are people who have never met before. They have to come up with team names. Our name when we play is always “the Ringers.” But we don’t compete with them, we demo the machine that we put together at the workshop.

Robot: Where can people go to find out more about the games that can be played with VEX kits?
Dan: People can visit Visualedgeinc.biz. That website is linked to Innovation First’s product page; Visual Edge games are sold exclusively through the Innovation First website. Go to Innovationfirst.com and Vexlabs.com, and click on the education button at the top. When you get to the education page you’ll see the Visual Edge banner on the right. You can also learn about the competitions by reading Robot magazine. We have a recommended reading list and Robot is actually at the top.

Robot: Are workshops on robotics curriculum on the radar of most states?
Dan: Initially, when we convinced the state to participate, there was no request for a proposal (RFP). Initially they did not have a clue, and this is true of a lot of states, I think. They don’t know how to fund such programs or even enough about them to produce an RFP. For many it’s a “club event” and not on the radar to be used to teach something. This is an early stage in this emerging arena, and the potential for expansion of robotics in education remains huge.

Robot: Are you extending these grant programs to other states?
Dan: Innovation First and others want to see this program spread to other states. IFI has put me in contact with people in California. People from Colorado, Tennessee, Kentucky and Texas have come to me and said we’d like to start this, can you tell us what is going on? I can duplicate this outside of Indiana, and we’ve kicked around some approaches. 
     If a state is willing to invest a small amount of money, I can come and train people in what we have used. They can opt to use it or not, that’s fine. I can give others the full run down on how we have been successful. In fact, any who wish to learn more about how they can start robotics programs for teachers in their state should contact me atdward@ivytech.edu.

Robot: Thank you Dan, for granting this interview. We wish you and your partners, continued success in bringing robotics into the classroom.
Dan: My pleasure!

GATEQUESTIONS

THEORY OF COMPUTATION

Q1. Choose the correct statement. The set of all strings over an alphabet S ={0,1} with the concatenation operator for strings a) does not form a group b) forms a noncommutative group c) does not have a right identity d) forms a group if the empty string is removed from S * Q2. Consider the set of all strings S * over an alphabet S ={a,b} with the concatenation operator for strings, and a) the set does forms semigroup b) the set does not form a group c) the set has a left and right identity d) the set forms a monoid Q3. Consider the set of all strings S * over the alphabet S ={a,b,c,d,e} with the concatenation operator for strings. a. the set has a right identity and forms a semigroup b. the set has a left identity and forms a monoid c. the set does not form a commutative group but has an identity d. the set does not form a semigroup with identity Q4. Nobody knows yet if P = NP. Consider the language L defined as follows: L=()+1)* if P = NP And L=j otherwise Which of the following statements is true? a) L is recursive b) L is recursively enumerable but not recursive c) L is not recursively enumerable d) Whether L is recursive or not will be known after we find out if P = NP Q5. Consider the language defined as follows L= {a^n b^n|n>=1} if P=NP And L={ww|w in (a+b)+} otherwise Which of the following statements is true? a) L is recursive but not context sensitive b) L is context sensitive but not context free c) L is context sensitive d) What L is will be known after we resolve the P=NP question Q6. Consider the language defined as follows L=(0+1)* if the CSLs are closed under complement And L=(0*1)*0* if P=NP And L=(10*)1* if P is not the same as NP Which of the following statements is true? a) L is always a regular set b) L does not exist c) L is recursive but not a regular set d) What L is will be known after the two open problems P = NP and the closure of CSLs under complement are resolved Q7. Consider the language defined as follows L=(0+1)* if man goes to Mars by 2020AD And L=0* if man never goes to the Mars Which of the following is true? a. L is context free language but not recursive b. L is recursive c. Whether L is recursive or not will be known in 2020AD d. L is a r.e. set that is not regular Q8. Given an arbitrary context free grammar G, we define L as below. L=(0+1)* if G is ambiguous And L=j if G is not ambiguous a. L is a context-free language b. L is recursive but not r.e. c. What L is depends on whether we can determine if G is ambiguous or not d. What L is is undecidable Q9. Given an arbitrary turing machine M and a string w we define L as below. L=(0*1)*0* if M halts on w And L=(0*1*)* if M does not halt on w a. The type of L is undecidable because of the halting problem b. L is a context-sensitive language c. L is recursively enumerable and not context-free d. L is context sensitive and not regular Q10. Consider the language L defined below L=(0+1)* if P=NP And L=(a^nb^n|n>=1} otherwise a. Whether L is a regular set that is not context-free will be known after we resolve the P=NP question. b. Whether L is context-free but not regular will be known after we resolve the P=NP question c. L is context-sensitive d. L is not recursive Q11. It is undecidable if two cfls L1 and L2 are equivalent. Consider two cfls L1 and L2 and a language defined as follows L={a^nb^nc^n|n>=234} if L1=L2 And L={a^nb^nc^nd^n|n>=678} otherwise a. L is recursive b. L is context-free c. We can never say anything about L as it is undecidable d. L is regular Q12. At present it is not known if NP is closed under complementation. Consider L defined as below L={w wR w| w in (0+1+2)* and wR is the reverse of w} if NP is closed under complement And L = {a^nb^nc^nd^ne^n|n>=34567} otherwise a) L is recursive b) L is context-free and not context-sensitive c) L is recursively enumerable but not recursive d) We will be able to say something about L only after we resolve the NP complementation issue Q14. Nobody knows if P=NP at present. Consider a language L as defined below L=(0+1)* if satisfiability is in P L=(0*1)0* if satisfiability is not in P L=(1*0)1* if 3-sat is in P L=(0*1*)* if 3-sat is not in P L=(0*1*0*1*)* if 0/1 knapsack problem is in P L=(1*0*1*0*)* if 0/1 knapsack problem is not in P L=(0*(00)*(1*11*)*) * if max-clique problem is in P L=(0*(00)*(1*11*)*) * if node-cover problem is not in P L=(0*1*)****(010)* if edge-cover problem is not in P L=(0* + 1* + (00)* + (11)*)*(0100101010)* if the chromatic problem is not in P What can we say about the string 0000111100001111=x a) x is always in L b) whether x is in L or not will be known after we resolve P=NP c) the definition of L is contradictory d) x can never be in L Q15. An arbitrary turing machine M will be given to you and we define a language L as follows L=(0+00)* if M accepts at least one string L=(0+00+000)* if M accepts at least two strings L=(0+00+000+0000)* if M accepts at least three strings --------- --------- L=(0+00+000+---+0^n) *if M accepts at least n-1 strings Choose the correct statement. a) We cannot say anything about L as the question of whether a turing machine accepts a string is undecidable b) L is context-sensitive but not regular c) L is context-free but not regular d) L is not a finite set Q16. We are given two context-free languages L1 and L2 and L defined as below a) L=(0+1)* if L1=L2 b) L=((0+00+000)*(1+11+111)*)* if L1 is contained in L2 c) L=((0(10)*)*(1(01)*)* if L2 is contained in L1 d) L=(00+11+0+1)*(0+00+000)* if L1 and L2 are incomparable a) As all the conditions relating to L1 and L2 are undecidable we cannot say anything about L b) L is recursively enumerable c) L is recursive but not context-sensitive d) L is context-sensitive but not context-free e) L is context-free but not regular Q17. It is undecidable if an arbitrary cfl is inherently ambiguous. We are given a cfg G and the language L is defined as below L= (0+1)*01(0+1)* U 1*0* if L(G) is inherently ambiguous L=(0+1)*10(0+1)* U 0*1* if L(G) is not inherently ambiguous Choose the incorrect statement a) L is regular b) L iscontext-free c) L is context-sensitive d) The above choices can be resolved only if we know if L(G) is inherently ambiguous or not Q18. We are given an arbitrary turing machine M and define the language L as below L=(0*+1*)* if M halts on blank tape L=(0+1*)* if M ever prints a 1 L=(0*+1)* if M ever enters a designated state q L=((0+1+00+11+000+111)+)* if M accepts an infinite set L=0*(10*)* if M accepts a finite set L=1*(01*)* if M accepts exactly 45 strings Choose the correct statement with reference to the string x=00000111111000000111111 a) x is in L b) x is not in L c) we can never decide if x is in L as all the problems of the turing machine are undecidable d) whether x is in L depends on the particular turing machine M Q19. We are given a language L defined as follows L=(0+1)* if the Hamiltonian circuit problem is in P L=(0*1*+0)* if the Traveling salesman problem is not in P L=(0*1*1)*0* if the bin packing problem is in P a) the definition of L is contradictory b) What L is will be known after we resolve the P=NP question c) L if a finite set d) The string 01010101001010110010101 is in L Q20. The intersection of two cfls can simulate a turing machine computation. We are given two cfls L1 and L2 and define the language L as below a) L=(00)* if the intersection of L1 and L2 is empty b) L=((0(00)*)(0(00)*))* if L1 is regular c) L=(00+0000+000000)* if L2 is not regular d) L=(00)*+(0000)* if the complement of L1 is a cfl a) L is a finite set b) L is a regular set c) L is undecidable d) L is recursive but not context-free Result Page:- 1-20 | 21-40 | 41-60 | 61-80 | 81-100 | 101-120 |

Ashtavaidyans

Ashtavaidyans are believed to be the traditional Ayurvedic physicians of Kerala and are from Namboothiri community. They are masters of the eight branches of medicine from which the word Ashtavaidyan is originated. They wrote several books incorporating their observations and clinical experiences. "Chikitsa Manjari", "Yogamithram", "Abhidhana Manjari", "Alathur Manipravalam", "Sindoora Manjari" and "Kairaly Commentary on Ashtanga Hridayam" are some of them. They come under the family of Vaagbhatachaaryan, one of the members of Brihat Trayee. Brihat Trayees are three authentic Aachaaryans, namely Susruthan, Charakan and Vaagbhatan.

According to Mr. N V K Varier's "Ayurveda Charithram", the word Ashtavaidyans does not refer to eight designated families of physicians, but rather to 18 Ashtaangavaidyans each one designated to 18 Sabhaamadhams (Vedam Schools) serving the many (32) Graamams of Kerala. These families were learned experts proficient in all the eight branches (Ashtaangams) of Ayurveda system (Poorna Vaidyans or complete physicians). The word Ashtaangavaidyans were later apparently reduced to Ashtavaidyans. It so happens that, in the absence of male members, several of these families had to be finally merged into eight of these families. The families are listed below with the existing families in the left column. Except Aalathiyoor and Kaarathol who are Nambis, all others are Moosses.

1.	Aalathiyoor Nambi	1. Aalathiyoor Nambi 2. Kaarathol Nambi 3. Choondal Mooss
2.	Elayidath Thaikkatt Mooss	4. Elayidath Thaikkatt Mooss 5. Kuriyedath Mooss (Njarakkal Mooss) 6. Kurumbempilly Mooss 7. Paduthol Mooss
3.	Pazhanellippurath Thaikkatt Mooss	8. Pazhanellippurath Thaikkatt Mooss 9. Peringavu Mooss 10. Parappur Mooss
4.	Kuttancherry Mooss	11. Kuttancherry Mooss 12. Vatuthala Mooss 13. Akalaanath Mooss
5.	Vayaskara Mooss	14. Vayaskara Mooss
6.	Chirattamon Mooss	15. Chirattamon Mooss (Olassa Mooss)
7.	Velluttu Mooss	16. Velluttu Mooss 17. Ubhayur Mooss
8.	Pulamanthol Mooss	18. Pulamanthol Mooss

Of these, Kaarathol Nambi either became extinct without any male children, or became Vaidyamadham. Moreover, there are no practising physicians in the families of Kuttancherry, Vayaskara and Velluttu Mooss, at present.

Another version is that it was Lord Parasuraman who brought Brahmanans (Namboothiris) to Kerala, assigned eight of the families as physicians, and these families came to be known as Ashtavaidyans. There is a third view which states that eight prominent disciples of Vaagbhata and their families continued the Ashtaangahridayam method of treatment, thus prompting the dual meaning of the word. Some believe that Vaagbhata came to Kerala and composed Ashtaangahridayam sitting on a rock near Thiruvizha temple, though historians contest this. Anyway, while the rest of the country follows Charaka and Sushrutha, Kerala follows Vaagbhata's Ashtaangahridayam, and this strict method of treatment is world-renowned.

As mentioned earlier, all the families are addressed as Moosses rather than Namboothiris, except Aalathiyoor and Kaarathol who are called Nambis. Ashtavaidyans are given a slightly depressed status perhaps because they have to examine dead bodies, perform surgical operations and use and follow Budhist Granthams (Treatises). However, considerable respect and place are given to them by the Namboothiri community.

Owing to the slightly lower status for Moosses, they are not permitted inside Yaaga saalaas, a place where Yaagams are performed. It is, however essential to have a physician nearby. This was assigned to Vaidyamadham. It is likely that Kaarathol Nambis were upgraded to Namboothiris and brought to Mezhathol for this purpose. Vaidyamadham was said to be the physician for all the 99 Yaagams performed by Mezhathol Agnihothri. The family follows Aalathiyoor Nambi's treatment methods, which points to the possibility of his ancestry to Kaarathol Nambi, who was himself trained under Aalathiyoor Nambi.

There are not many historical studies nor records documenting this rich heritage. Their knowledge, ideas, experiences and ideals will be of great value not only to the present generation, but also to the future ones to come. Kerala is in a way fortunate to have had a number of people taught and trained by the Ashtavaidyans. The heritage has even transgressed to other communities and religious faiths.

Saalaavaidyan

Aayurvedam had developed along two scientific streams - "Sasthrakriya" (surgery) and "Chikilsa" (treatment). The surgeons are called "Dhanwanthareeyans" and the physicians, "Bhaaradwaajeeyans". The luminaries of that period were Susruthan and Charakan. Since treatises, "Susrutha samhitha" and "Charaka samhitha" are observed to have been revised, it may have to be surmised that what we see today are not the original Granthams.

Since those days, it was the great Vaagbhataachaaryan who tried to rejuvenate and modernise Aayurvedam. It must be mentioned that Vaagbhataachaaryan was a liberal, when one looks at his life and works. His works reveal his intention to integrate the two branches. He composed his treatises in a meticulous and orderly manner, choosing apt words, and with even a poetic ring to them. Unfortunately, he had to suffer a lowering of his social status (a minor defilement) throughout his life for his Budhist beliefs. Consequently, most people did not accept his Granthham. During that period, when he reached Kerala, he was given a very warm and hearty welcome. That made him happy and it is said that he spent rest of his life here in Kerala. He is rightly considered as the "Aachaaryan" of the Aayurvedic tradition in Kerala. His work "Ashtaanga hrudayam" describes the eight branches and , also led to the popular "Ashtavaidyam" ( Click Here for "Aayurvedam"). The full form of this term should have been "Ashtangaayurveda vaidyam" (the eight branches of the medical/ health sciences). One has to view the Kerala situation which existed at that time.

It must be surmised that during that period, just as there was a dearth of an effective health science, there was also a dearth of Vedic Karmams and Yajnams. As if to fill that void, was born the great Mezhathol Agnihothri ( Click Here ). It must have been due to his singular efforts that the Yajnam culture was rejuvenated in this area. Vaagbhatan must have been born before Agnihothri, because, when Agnihothri started Yaagams, the hereditary practice of Ashtaangaayurvedam was already prevalent here. There are clear indications of the essentiality of a physician in Yaagasaala, (the place where Yaagam is performed) to take care of the medical problems of the performers ("Rithwiks") even during the Vedic period. It is believed that in those days, the "Aswineedevans" were given the physicians' lower grade ("Paathithyam") and prevented from entering the Yaagasaala, but then, sage Chyavanan, through his blessings, is presumed to have removed this problem. The same must also have happened in Kerala during the revival of Vedic culture.

The Ashtaanga Aayurveda doctors of Kerala, who follow the Vaagbhata school used to practice both streams - surgery and treatment, but the lowering of the grade was assigned only for the surgeons. Thus it became necessary to find an individual or a family of physicians to be assigned to the Yaagasaala needs. That is how and when the Vaidyamadham family of Mezhathur was honoured with this task, selected perhaps from one of the several families of the Vaagbhata tradition.

This assignment may not have been to an individual, since there is a 300-plus published and unpublished palm leaf Granthham collection pointing to the hereditary tradition of the "Vaidyamadham swaroopam". Such a huge ancestral collection would not possibly have been there if it were assigned to an individual. Historically, there were 18 families with Ashtavaidyam tradition, but many became extinct ( Click Here for Ashtavaidyans). Kaarathol Nambi was one such, and lived somewhere close to Aalathur Nambi and related to them. Considering the many commonalities in the treatment techniques of Vaidyamadham and Kaarathol Nambi, some believe that the latter was inducted as the Yaagasaala doctor through the efforts of Agnihothri and was resettled at Mezhathur. These conclusions are, at best, only logical conjectures, and beyond solid proof.

There is another possibility. Vaagbhataachaaryan apparently had two great disciples, one with pen name "Indu" and the other "Jarjatan", as mentioned in the Vaagbhata invocation song ("Dhyaana slokam"). Vaidyamadham may have their ancestry in the family of the greater one of the two, Indu. This surmise is derived from the fact that the two of the three palm-leaf copies of the "Vyaakhyaanams" (explanations) of "Ashtaangahrudayam" and "Sangraham" (summary) written by Indu were in the Vaidyamadham collection. One of these two copies was taken by an eminent and renowned member of the family known as Kunchu Apphan, some 130 years ago (say, around AD 1870) to Edappally where he was staying as physician to the Kochi Royal family. He used it for reference when he had to teach "Vaidyam" to students. It was apparently lost after this time. The other copy is still in the family collection. So much is the background of Vaidyamadham family.

Today, Vaidyamadham swaroopam is the only family in Kerala with the Bhaaradwaajeeya tradition. They are not permitted to do surgery ("Salyasaalaakya" or "Sasthrakriya") that causes "paathithyam". This was perhaps the reason for their Vedic rights and assignment as "Saalaavaidyan". The normal practice in Yajna culture is for the Yajamaanan (master, the person actually doing the Yajnam) to consult and get the permission from the Rithwiks and the Vaidyan before deciding on the Yaagam. Once decided, more than one person requests the Raja of Kollangode for the "Soma" ( Click Here ) and the leather. He is called the "Gandharvan" who protects the Soma. The age-old rule is for the Saalaavaidyan to be always present in the Saala from the beginning to the end of the Yaagam, for looking after the health and medical needs of the Yajamaanan and Rithwiks, as they are not permitted to leave the premises to the end. Till today the formality continues, though he may not be present always and every day.

The Vaidyan's position in the Yaagasaala is in the area called "Ulkkaram". He is the only person who is provided with an "Aavanappalaka" (a special low wooden seat) to sit on. If, for any reason (say "Pula" or defilement), he cannot be present, he usually sends a replacement. In his absence, the standard fee ("Prathipphalam") of 16 Panam is kept on his seat. It is of special interest to note that while the "Paradevatha" of all other Ashtavaidya families is Dhanwanthari, Vaidyamadham's Kuladevatha" is Dakshinaamoorthy (form of Lord Siva in meditation).

Vaidyamadham is supposed to participate as Saalaavaidyan in any Yaagam performed in Kerala, and this has so far been adhered to. In Sukapuram and Perumanam Graamams there used to be a practice of giving a share to the Karmis. It is done in any one of the eight days (only three in Perumanam temple) from Chithra to Uthraatam star of Medom (Malayalam month - mid-April to mid-June) after a bath in the temple tank followed by worship. The share is called "Pazhuthi". At Sukapuram the traditional way of wearing the cloth ("Thattudukkal") is necessary before worshipping in the temple.

Srinivasa Ramanujan

It is one of the most romantic stories in the history of mathematics: in 1913, the English mathematician G. H. Hardy received a strange letter from an unknown clerk in Madras, India. The ten-page letter contained about 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory (Here is a .dvi file with a sample of these results). Every prominent mathematician gets letters from cranks, and at first glance Hardy no doubt put this letter in that class. But something about the formulas made him take a second look, and show it to his collaborator J. E. Littlewood. After a few hours, they concluded that the results "must be true because, if they were not true, no one would have had the imagination to invent them".

Thus was Srinivasa Ramanujan (1887-1920) introduced to the mathematical world. Born in South India, Ramanujan was a promising student, winning academic prizes in high school. But at age 16 his life took a decisive turn after he obtained a book titled A Synopsis of Elementary Results in Pure and Applied Mathematics. The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. It was in no sense a mathematical classic; rather, it was written as an aid to coaching English mathematics students facing the notoriously difficult Tripos examination, which involved a great deal of wholesale memorization. But in Ramanujan it inspired a burst of feverish mathematical activity, as he worked through the book's results and beyond. Unfortunately, his total immersion in mathematics was disastrous for Ramanujan's academic career: ignoring all his other subjects, he repeatedly failed his college exams.

As a college dropout from a poor family, Ramanujan's position was precarious. He lived off the charity of friends, filling notebooks with mathematical discoveries and seeking patrons to support his work. Finally he met with modest success when the Indian mathematician Ramachandra Rao provided him with first a modest subsidy, and later a clerkship at the Madras Port Trust. During this period Ramanujan had his first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society. Still no one was quite sure if Ramanujan was a real genius or a crank. With the encouragement of friends, he wrote to mathematicians in Cambridge seeking validation of his work. Twice he wrote with no response; on the third try, he found Hardy.

Hardy wrote enthusiastically back to Ramanujan, and Hardy's stamp of approval improved Ramanujan's status almost immediately. Ramanujan was named a research scholar at the University of Madras, receiving double his clerk's salary and required only to submit quarterly reports on his work. But Hardy was determined that Ramanujan be brought to England. Ramanujan's mother resisted at first--high-caste Indians shunned travel to foreign lands--but finally gave in, ostensibly after a vision. In March 1914, Ramanujan boarded a steamer for England.

Ramanujan's arrival at Cambridge was the beginning of a very successful five-year collaboration with Hardy. In some ways the two made an odd pair: Hardy was a great exponent of rigor in analysis, while Ramanujan's results were (as Hardy put it) "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account". Hardy did his best to fill in the gaps in Ramanujan's education without discouraging him. He was amazed by Ramanujan's uncanny formal intuition in manipulating infinite series, continued fractions, and the like: "I have never met his equal, and can compare him only with Euler or Jacobi."

One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) of partitions of a number n. A partition of a positive integer n is just an expression for n as a sum of positive integers, regardless of order. Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. The problem of finding p(n) was studied by Euler, who found a formula for the generating function of p(n) (that is, for the infinite series whose nth term is p(n)xⁿ). While this allows one to calculate p(n) recursively, it doesn't lead to an explicit formula. Hardy and Ramanujan came up with such a formula (though they only proved it works asymptotically; Rademacher proved it gives the exact value of p(n)).

Ramanujan's years in England were mathematically productive, and he gained the recognition he hoped for. Cambridge granted him a Bachelor of Science degree "by research" in 1916, and he was elected a Fellow of the Royal Society (the first Indian to be so honored) in 1918. But the alien climate and culture took a toll on his health. Ramanujan had always lived in a tropical climate and had his mother (later his wife) to cook for him: now he faced the English winter, and he had to do all his own cooking to adhere to his caste's strict dietary rules. Wartime shortages only made things worse. In 1917 he was hospitalized, his doctors fearing for his life. By late 1918 his health had improved; he returned to India in 1919. But his health failed again, and he died the next year.

Besides his published work, Ramanujan left behind several notebooks, which have been the object of much study. The English mathematician G. N. Watson wrote a long series of papers about them. More recently the American mathematician Bruce C. Berndt has written a multi-volume study of the notebooks. In 1997 The Ramanujan Journal was launched to publish work "in areas of mathematics influenced by Ramanujan".

Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Ramanujan was born in his grandmother's house in Erode, a small village about 400 km southwest of Madras. When Ramanujan was a year old his mother took him to the town of Kumbakonam, about 160 km nearer Madras. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. In December 1889 he contracted smallpox.

When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.

It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19^th century. The book, published in 1856, was of course well out of date by the time Ramanujan used it.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery.

Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of Madras. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.

In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa's College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr's book.

Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl S Janaki Ammal. Ramanujan did not live with his wife, however, until she was twelve years old.

Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.

In 1911 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General's Office in Madras. It was then suggested that he approach Ramachandra Rao who was a Collector at Nellore. Ramachandra Rao was a founder member of the Indian Mathematical Society who had helped start the mathematics library. He writes in [J.%20Indian%20Math.%20Soc.%2012%20(1920),%2087-90.',30)" onmouseover="window.status='Click to see reference';return true">30]:-

A short uncouth figure, stout, unshaven, not over clean, with one conspicuous feature-shining eyes- walked in with a frayed notebook under his arm. He was miserably poor. ... He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. ... I asked him what he wanted. He said he wanted a pittance to live on so that he might pursue his researches.

Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust. In his letter of application he wrote [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]:-

I have passed the Matriculation Examination and studied up to the First Arts but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject.

Despite the fact that he had no university education, Ramanujan was clearly well known to the university mathematicians in Madras for, with his letter of application, Ramanujan included a reference from E W Middlemast who was the Professor of Mathematics at The Presidency College in Madras. Middlemast, a graduate of St John's College, Cambridge, wrote [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]:-

I can strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work.

On the strength of the recommendation Ramanujan was appointed to the post of clerk and began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people working round him with a training in mathematics. In fact the Chief Accountant for the Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the distribution of primes in 1913 on Ramanujan's work. The professor of civil engineering at the Madras Engineering College C L T Griffith was also interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers.

Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E W Hobson and H F Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. In Ramanujan's letter to Hardy he introduced himself and his work [Ramanujan%20:%20Am%20inspiration%202%20Vols.%20(Madras,%201968).',10)" onmouseover="window.status='Click to see reference';return true">10]:-

I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.

Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. On 8 February he replied to Ramanujan [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3], the letter beginning:-

I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes:
(1) there are a number of results that are already known, or easily deducible from known theorems;
(2) there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance;
(3) there are results which appear to be new and important...

Ramanujan was delighted with Hardy's reply and when he wrote again he said [Collected%20Papers%20(Cambridge,%201927).',8)" onmouseover="window.status='Click to see reference';return true">8]:-

I have found a friend in you who views my labours sympathetically. ... I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government.

Indeed the University of Madras did give Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy's at Trinity College and who met with Ramanujan while lecturing in India.

Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April 1914 and was met by Neville. After four days in London they went to Cambridge and Ramanujan spent a couple of weeks in Neville's home before moving into rooms in Trinity College on 30^th April. Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.

Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education. He wrote [Dictionary%20of%20Scientific%20Biography%20(New%20York%201970-1990).',1)" onmouseover="window.status='Click to see reference';return true">1]:-

What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity.

Littlewood was asked to help teach Ramanujan rigorous mathematical methods. However he said ([Minerva%2029%20(1991),%20393-419.',31)" onmouseover="window.status='Click to see reference';return true">31]):-

... that it was extremely difficult because every time some matter, which it was thought that Ramanujan needed to know, was mentioned, Ramanujan's response was an avalanche of original ideas which made it almost impossible for Littlewood to persist in his original intention.

The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.

Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes. In February 1918 Hardy wrote (see [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]):-

Batty Shaw found out, what other doctors did not know, that he had undergone an operation about four years ago. His worst theory was that this had really been for the removal of a malignant growth, wrongly diagnosed. In view of the fact that Ramanujan is no worse than six months ago, he has now abandoned this theory - the other doctors never gave it any support. Tubercle has been the provisionally accepted theory, apart from this, since the original idea of gastric ulcer was given up. ... Like all Indians he is fatalistic, and it is terribly hard to get him to take care of himself.

On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians, namely Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and Whitehead. His election as a fellow of the Royal Society was confirmed on 2 May 1918, then on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.

The honours which were bestowed on Ramanujan seemed to help his health improve a little and he renewed his effors at producing mathematics. By the end of November 1918 Ramanujan's health had greatly improved. Hardy wrote in a letter [Ramanujan%20:%20Letters%20and%20commentary%20(Providence,%20Rhode%20Island,%201995).',3)" onmouseover="window.status='Click to see reference';return true">3]:-

I think we may now hope that he has turned to corner, and is on the road to a real recovery. His temperature has ceased to be irregular, and he has gained nearly a stone in weight. ... There has never been any sign of any diminuation in his extraordinary mathematical talents. He has produced less, naturally, during his illness but the quality has been the same. ....

He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty has never been affected in the least by success - indeed all that is wanted is to get him to realise that he really is a success.

Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died there the following year.

The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function. On the other hand he had only a vague idea of what constitutes a mathematical proof. Despite many brilliant results, some of his theorems on prime numbers were completely wrong.

Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan's death.

In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and this was later proved by Rademacher.

Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.

The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75^th anniversary of his birth.