SaiK: now you can see the one in red can be written as followsmatrimc wrote: 13a = 39 n + 2 x
where
3a - 2b = 13 n
2a + 3b = x
a an integer
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+--- ---+
| 3 -2 | = A
| 2 3 |
+--- ---+
and the columns of A are orthogonal. So when we A' A is the diagonal diag(13,13).
So the solution of A' A x = A' b should lie in the field F_13^2 as A' A contains the unit of F_13^2.
For a career in mathematics
http://www.thehindu.com/sci-tech/scienc ... 747091.ece
S. Kesavan
In India a mathematical career has been regarded as being synonymous with a teaching career, but a trained mathematician can be very well employed outside academia. Photo: S.R. Raghunathan
It is indeed possible to build a perfectly satisfying career in mathematics if one is deeply interested in the subject
India has a long and ancient mathematical tradition. The Sulvasutras, Vedic texts for the construction of ritual altars, contain a lot of geometrical results and constructions. These include a statement of the Pythagoras Theorem, an approximation to the value of ‘pi', and the ratio of the circumference of a circle to its diameter. India gave the world the decimal place value system, the modern way of writing numbers, and above all, the number ‘zero.' It boasts of mathematical schools like those of Aryabhata and Bhaskara. Much later, in the 15th century, came the flourishing School of Madhava in Kerala, which anticipated, by more than 200 years, several results of the Calculus invented by Newton and Liebniz.
There was a complete break in this tradition during the years of colonial rule. In the 20th century, perhaps inspired by Ramanujan's life, there was a revival, especially in the south, of mathematical research. In the post-Independence era, the Government of India established some schools of excellence, where several individuals distinguished themselves, and continue to distinguish themselves, by doing excellent work.
Nevertheless, for a country of India's size, despite having a large scientific workforce, we have failed to make the kind of international impact that countries like, say, China, have made. India's own scientific leaders have often bemoaned the ‘ocean of mediocrity' that has been created.
The main problem is that a mathematical career has been regarded as being synonymous with a teaching career. We religiously teach our children slokas like Guru Brahma, Guru Vishnu, Guru Devo Maheshwarah. However, equally cruelly and callously we say things like vakkillathavanukku vathiyar velai (a teacher's vocation is for those who have no other option). This has become a self-fulfilling prophecy of sorts. Barring a minuscule number of exceptions, India's brightest minds are not engaged in scientific research. The situation in general is that those who fail to join professional courses leading to gainful employment come to research as a last resort. These are the ones who will become the (uninspiring) teachers of the future — and we are caught in a vicious cycle.
The situation should, in reality, be the opposite. Those taking to a research career should be those who are passionately involved in the subject. As the experience of the information technology industry shows, anybody with a reasonable degree can be trained on the job and be well-employed, whereas that is not the case in academia.
It is indeed possible to build a perfectly satisfying career in mathematics (and much of this applies to other pure sciences as well) if one is deeply interested in the subject.
Job scene
Look at the job scene. A trained mathematician can be very well employed outside academia. Government departments engaged in space research (the Indian Space Research Organisation, or ISRO), defence research (Defence Research and Development Organisation, or DRDO), aeronautical research (National Aeronautics Limited, or NAL), all employ mathematicians to solve their special problems. Today, cryptology is in vogue (the systems ensuring the safety of your credit card transactions are based on some very sophisticated mathematics). Organisations such as the DRDO and the Society for Electronic Transactions and Security (SETS) are interested in mathematicians with training in this area. Financial mathematics is another area that leads to well-paid jobs. Computer giants such as IBM and Microsoft have research departments which have highly paid scientists who are either mathematicians or theoretical computer scientists. (They can, for all practical purposes, be considered as mathematicians). Thus, there is plenty of scope, outside academia, for well-paid jobs for mathematicians.
Having said this, it must be emphasised that the majority of mathematicians will end up in academic jobs, namely, in research and teaching.
What are the plus points of such a vocation?
•In India, all these jobs are in universities or in public-funded research institutions. With the implementation of the recommendations of the Sixth Pay Commission, the salary is nothing to be sniffed at. The entry point (roughly between the ages of 28 and 32) is that of an Assistant Professor, who can expect to start at a monthly basic salary of Rs.30,000. To this, add the dearness allowance (which has well crossed 502 per cent of the basic), transport allowance, and (if accommodation is not provided by the employer) a house rent allowance (which touches 30 per cent of the basic in the metros). Thus, before tax, we arrive at something like Rs.50,000 or more a month. This, unlike in industry, is not the ‘cost to company' but what the employee actually gets. Added to this are perquisites such as comprehensive health care, leave travel concession, aid to children's education and employer's contribution to the provident fund or the pension fund. All in all, the remuneration today does guarantee a very good standard of living with all the creature comforts.
In order to attract young Ph.D.s who have done rather well by way of research, especially but not limited to those from abroad who seek employment in India, the Department of Science and Technology (DST) offers the Ramanujan Fellowship for three years. It carries a high salary and a generous contingency grant that allows purchase of research equipment, travels abroad for conferences, and so on. Institutions like the IITs and the Indian Institute of Science (IISc) in Bangalore also offer generous start-up grants to freshly-recruited faculty members to facilitate their research.
•Job security.
•Job satisfaction: you get to choose your research problems.
•A good quality of life: the timings are regular with vacation periods that are well-defined.
•Plenty of opportunities to set up research collaborations with fellow-researchers in India and abroad, providing possibilities of interesting domestic and international travel.
•Being in contact with young minds all the time has a rejuvenating effect on one's outlook to life.
On the other hand, one should ensure that one is really interested in the subject. To rise in the profession one needs to have a reasonably steady research output for nearly three to four decades. The real downside is that the gestation and apprenticeship period is quite long. It takes about five years to get a master's degree and between three to five years more for the doctoral degree. Even after that, it is expected that a person does at least two years of post-doctoral work, which is the time when one emerges from the shadows of the thesis supervisor and chalks out one's own path of research. Thus, as mentioned earlier, one can expect to get one's first job when in the 28-32 age group. But this period is not financially barren, and the remuneration keeps increasing.
Job opportunities
What about job opportunities in India? There are three kinds of institutions of higher learning.
Purely research-oriented institutions like the Tata Institute of Fundamental Research (TIFR) in Mumbai, the Institute of Mathematical Sciences (IMSc) in Chennai, and the Harish Chandra Research Institute (HRI) in Allahabad. Interestingly, all these are autonomous aided institutions that are fully supported by the Department of Atomic Energy (DAE) of the Government of India. TIFR is now a deemed university, while the IMSc and the HRI are affiliated to the deemed university called the Homi Bhabha National Institute (HBNI) that covers all other aided institutions of the DAE.
Institutions of teaching and research which can offer degrees but do not come under the purview of the University Grants Commission (UGC). These are set up by Acts of Parliament, and some come under the Ministry of Human Resource Development (MHRD). These are the Indian Statistical Institute (ISI), the IITs, the IISc, and the newly set up Indian Institutes of Science Education and Research (IISERs) in Bhopal, Kolkata, Mohali, Pune and Thiruvananthapuram, and the National Institute of Science Education and Research (NISER) in Bhubaneswar (set up by the DAE). Then there is the precursor to these latter new institutes, the unique Chennai Mathematical Institute (CMI), which is an example of public-private partnership. ISRO has also established its own such institution in Thiruvananthapuram.
The State and Central universities. While the State universities have plenty of vacancies, these being filled is often tied to the policies and politics of the State governments.
All the other institutions of research and teaching mentioned above have well-established and transparent methods of selection, and all of them have a crying need for fresh faculty. In fact, the need is so great and the supply so meagre that the age of retirement has been increased to 65 for these institutions. And in many cases they are allowed to re-employ superannuated faculty members till they are 70. The government has suddenly started NISER, the five IISERs and about eight new IITs, all of which need faculty members. These are currently functioning with a bare minimum of recruits, augmented by adjunct faculty members, who are retired mathematicians. This is not sustainable in the long run.
Thus, for those who hold a reasonably good doctoral degree, there are plenty of job opportunities in such institutions. This will be so for a long time to come.
Even the existing institutions like the IITs face continuous attrition due to retirement of faculty members who were engaged from the 1960s onwards.
Training
Now for the training process of a mathematician in India. The regular route for a student is a three-year B.Sc. course followed by a two-year M.Sc. programme in mathematics, after which she or he could join a doctoral programme in a recognised university or research institution. There are the following variants to this theme.
The IIT-Kanpur pioneered the five-year M.Sc. programme (admission is through the joint entrance examination) which combined the B.Sc. and M.Sc. programmes. IIT-Bombay followed suit. Now, this pattern is followed by all the IISERs and NISER. The Central University of Hyderabad and that of Pondicherry have also started such programmes.
Recently, the three science academies in India have been advocating educational reform that involves the introduction of a four-year B.S. programme followed by a year of research and training leading to an M.S. The IISc will launch the first such programme in August 2012.
Institutions of pure research (the TIFR, the IMSc and the HRI), the IISc and the CMI also have integrated Ph.D. programmes. Promising students are selected after a bachelor's degree in any science discipline or engineering directly for their Ph.D. programmes, provided they clear the (very rigorous) entrance tests and interviews on a par with M.Sc. candidates. They pick up an M.Sc. degree after two initial years of course work and research.
All the IITs and universities also have independent M.Sc. and Ph.D. programmes. Admission is based on entrance tests and/or interview. The CMI has an M.Sc. programme in applications of mathematics with specialisation in financial mathematics and computational applications of mathematics. It is contemplating a stream specialising in cryptology. The ISI has an M. Math. Programme, held alternatively at its Kolkata and Bangalore campuses.
A special word on the undergraduate programmes of the CMI, which is B.Sc.(Hons.) in Mathematics and Computer Science in Chennai, and the ISI — B.Math at its Bangalore campus. These are not for the faint-hearted. But if a student has a strong taste and talent for mathematics from an early age, these are the places to go for mathematics education. Both these programmes are very intense. At the end of three years, the students can compete with any master's level student anywhere on equal terms — and often they fare better.
As a measure of the success of these programmes, it must be said that their graduates have managed to breach the U.S. firewall that requires a four-year collegiate-level education to enter graduate school, by being directly admitted, with full aid, to graduate schools such as Caltech, Chicago, Princeton, MIT (and so on in the U.S.), the Max Planck institutes in Germany, and the elite Ecole Normale Sup´erieure in France, after finishing the three-year degree programme. Students from the early batches have started completing their doctorate work and are already making a mark. It is gratifying that some have come back to take up positions in India.
Finally, about scholarships. First of all, there is the Kishore Vaigyanik Pratsohan Yojana which conducts a test for high school students. The successful ones opting for a career in science get a handsome scholarship all through their higher education, up to completion of the doctoral programme. The CMI and the ISI provide modest stipends to their undergraduates and postgraduates, together with tuition waiver, as long as the students maintain a healthy academic performance.
For the doctoral programmes, university students need to take an examination conducted by bodies like the Council of Scientific and Industrial Research or the DST for a research fellowship. The current rates are Rs.16,000 for the first two years and, subject to satisfactory performance, Rs.18,000 a month thereafter. There is an annual contingency grant as well.
All research institutions and institutions of teaching and research mentioned here have their own funding for Ph.D. scholarships at the same rates. In case the institution cannot provide subsidised accommodation on campus, house rent allowance at the same rates as applicable to faculty members is allowed.
Post-doctoral fellowships provide for a consolidated pay ranging from Rs.21,000 to Rs. 25,000 a month (with the provision for HRA), along with a contingency grant, depending on the candidate's post-doctoral experience.
The National Board for Higher Mathematics (NBHM), set up by the DAE to promote mathematics, conducts an examination every year for the award of a scholarship for M.Sc. programmes in mathematics in any recognised university or institution, and pays a monthly stipend of Rs.6,000. The advertisement appears in newspapers by the end of June; the written test is usually held towards the end of September. It also awards Ph.D. scholarships, at the same rates as other research fellowships, by conducting another examination which is advertised in November; the test is usually by the end of January or early February. The NBHM also offers post-doctoral fellowships.
To sum up, if a student has the taste and the talent for mathematics, it is possible to make a satisfying, interesting, respectable and remunerative career out of it. If you think you have it in you, just go for it. Study abroad if you really want to; it can broaden your horizons. But do come back to inspire future generations so that India will become a mathematical superpower in the coming decades.
Parents ought to let children do whatever they are best suited for — literature, dramatics, mathematics, painting and so on. They should not try to live out their ambitions through them. While it may be a status symbol to count a non-resident Indian in the family, as one grows older there is a pleasure and sense of security in having one's children living and working close by.
(The author is a Professor of Mathematics at the Institute of Mathematical Sciences, Chennai.)
Keywords: Srinivasa Ramanujan, Ramanujan birth anniversary, mathematics as career
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X - 4X = 1 + 2 + 3 + 4 + 5 + 6 + .....
-4 -8 -12 ...
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X - 4X = 1 - 2 + 3 - 4 + 5 - 6 +....
http://scitation.aip.org/content/aip/ma ... /PT.5.8029
The article is a very interesting read...In 2002, two men savagely attacked Jason Padgett outside a karaoke bar, leaving him with a severe concussion and post-traumatic stress disorder. But the incident also turned Padgett into a mathematical genius who sees the world through the lens of geometry.
Padgett, a furniture salesman from Tacoma, Washington, who had very little interest in academics, developed the ability to visualize complex mathematical objects and physics concepts intuitively. The injury, while devastating, seems to have unlocked part of his brain that makes everything in his world appear to have a mathematical structure.
FWIW: I read that article, and other pieces as there has been lot of media attention on him. I remain unimpressed as I have not seen a single point which shows any evidence of "genious" in that person. To me, it seems like people are very gullible when it comes to math, and even some mumbo-jumbo can make a news story. (eg "He started sketching circles made of overlapping triangles, which helped him understand the concept of pi, the ratio of a circle's circumference to its diameter. There's no such thing as a perfect circle, he said, which he knows because he can always see the edges of a polygon that approximates the circle...(so? - what math concept is here? - or talking about "Planck length" - which involves nothing more than repeating something read from wiki...)ArmenT wrote:A Beautiful Mind: Brain Injury Turns Man Into Math Genius
....
The article is a very interesting read...
Stop the watch when you answer the question..The smallest integer of a set of consecutive integers is -32. If the sum of these integers is 67, how many integers are in the set?
The following was interesting too, as this was answered by fewest number (less than 6% answered it correctly) of students..A circle passes through two diagonally opposite vertices of a 3-inch by 4-inch rectangle. What is the least possible distance between the center of the circle and a vertex of the rectangle?
Larry tells Mary and Jerry that he is thinking of two consecutive integers from 1 to 10. He tells Mary one of the numbers, and he tells Jerry the other number. Then the following conversation occurs between Mary and Jerry:
Mary: I don't know your number.
Jerry: I don't know your number, either.
Mary: Ah, now I know your number.
Assuming both Mary and Jerry used correct logic, what is the sum of the possible numbers Mary could have?
Actually it is -n.Multatuli wrote:My Guess would be -32. Don't ask for a systematic approach to the solution, I am just "reverse engineering" the final Mathcounts question.
A "systematic approach" would be try out -2, -3 etc.
Added later:
Just write a nice little proggie to do it.
5 or 17?Amber G. wrote: The following was interesting too, as this was answered by fewest number (less than 6% answered it correctly) of students..
(Mind you these are top (among the brightest) about 200 students nation wide - selected from about a few hundred thousand students) - you have about a minute to solve)
Larry tells Mary and Jerry that he is thinking of two consecutive integers from 1 to 10. He tells Mary one of the numbers, and he tells Jerry the other number. Then the following conversation occurs between Mary and Jerry:
Mary: I don't know your number.
Jerry: I don't know your number, either.
Mary: Ah, now I know your number.
Assuming both Mary and Jerry used correct logic, what is the sum of the possible numbers Mary could have?
25Amber G. wrote:Okay how about this.. From the same competition... (6th-8th grade students)
The following was interesting too, as this was answered by fewest number (less than 6% answered it correctly) of students..A circle passes through two diagonally opposite vertices of a 3-inch by 4-inch rectangle. What is the least possible distance between the center of the circle and a vertex of the rectangle?
(Mind you these are top (among the brightest) about 200 students nation wide - selected from about a few hundred thousand students) - you have about a minute to solve)
Larry tells Mary and Jerry that he is thinking of two consecutive integers from 1 to 10. He tells Mary one of the numbers, and he tells Jerry the other number. Then the following conversation occurs between Mary and Jerry:
Mary: I don't know your number.
Jerry: I don't know your number, either.
Mary: Ah, now I know your number.
Assuming both Mary and Jerry used correct logic, what is the sum of the possible numbers Mary could have?
I think both answers are incorrect..matrimc wrote:Amberg 2.5 and 11?
My son was in the school team. They did reasonably well but could not make it to the state.
(at least the official answer )Two random integers, a and b, are independently chosen, with replacement,
from 1 to 1000, inclusive. What is the probability that both 2^a + 2^b and 3^a + 3^b are multiples of 5?
(I got a very nice letter BTW, and glad that many young kids are interested in math)I forgot to mention. We had the exact same triangle challenge problem you gave to us in our team round. One of us even remember the correct solution and we got the answer right.
Good, this way is not too bad (and fast enough) but the answer is exactly 0.7Multatuli wrote:AmberG, I did it with analytical geometry, and the minimal distance is the distance between the points:
D=(0,4) and F=(-21/50,86/25), that is 0,69 or 0,7.
(Rectangle with vertices A=(0,0), B=(3,0), C=(3,4), D=(0,4), the circle passes through A and C, so the center of all the circles must lie on y=-3*x/4 + 25/8.)
.I know that the second is probably incorrect, so please explain the reasoning in detail, otherwise I just won't get it
