Five things to do as a graduate student in Mathematics

The following are the views of Mr. Mohammed Kaabar posted on AMS Graduate Blog just today:

I would like to share with you my first year experience as a graduate student in mathematics at Washington State University, and I want to give you some suggestions about what you should do as a graduate student in mathematics?. In Spring 2015, I started my first semester as a Ph.D student in Applied Mathematics, and during that semester, I wrote two math textbooks in differential equations and linear algebra, and I also gave three seminar’s talks in Applied Mathematics, as well as, I participated as invited technical program committee (TPC) member and invited reviewer for many international conferences and journals in applied math, physics, electrical engineering, and computer engineering. Most of these conferences published their accepted papers in major trade peer-reviewed publishing companies such as Springer and IEEE Xplore. Therefore, the following is a list of five different things that I highly recommend you to do as a graduate student in mathematics:

  • Join professional organizations in mathematics and other related fields: When I started my graduate studies in mathematics, I joined the Society for Industrial and Applied Mathematics (SIAM). It is easy to become a student member in SIAM because they offer a free membership for graduate students in some universities. My university was one of them, so I got a free membership. There are several math associations and societies such as American Mathematical Society (AMS) and Mathematical Association of America (MAA) that can provide you with good discounts on the prices of their student memberships.
  • Create a professional website: If you are a newly admitted graduate student in mathematics, I recommend you to create a professional website that includes your research interests, curriculum vitae, work experience, and courses you are currently teach. The advantage of having your own professional website is that many people will contact you by your website email to invite you as technical program committee (TPC) member, reviewer, editor, math team member for conferences and journals in mathematics and applied sciences. If you are going to teach a class, it is a good idea to add a section in your professional website that contains your lecture notes, solutions to your class assignments and quizzes, and study guides for exams.
  • Teach a course you like to teach: If you have been offered a teaching assistantship position at your department, I believe that most universities give you the option to request the courses you like to teach. Therefore, I recommend you to choose courses that interest you more than others because if you like the course you teach, your students will more likely appreciate the way you teach.
  • Participate in research groups: If you are a new graduate student in your department, I recommend you to contact your department chair, coordinator, advisor, and graduate studies chair to ask them about any available research groups to join them as a member so you can participate in the group’s research publications and seminar’s talks.
  • Participate in extra-curricular and academic related activities: When you start you graduate studies in mathematics, you will have a stress of work and study load. So, what you should do to relief this stress?. The answer is simple; many universities and colleges have student’s associations and clubs such as graduate student association and peer leadership program. For example, when I was student at Washington State University (WSU) and American University of Sharjah (AUS), I was an active member in peer leadership program, and I had also taken part taken part in competitions such as The International Electronics Synopsys Competition.

In conclusion, from the fifth point I mentioned above, I would like to focus on one case which I consider a great achievement in my work as a peer leader. One day, while I was walking along the corridors of AUS, I saw a student wondering, knowing neither where to go nor what to do. That student was as perplexed as a person going astray in the desert without being able to decide his direction. I approached him and asked him what he wanted. He told me that it was his first day at AUS and he did not know where and how to start. I assumed him that everything would be alright. Then, that student released a sigh of relief exactly the same feeling of our friend in the desert when a plane out of the blue sky took him out of the mire. That freshmen student was like a ship in a rough sea beaten by high waves, sometimes taking it west and some other times right. Imagine what would that person feel when he suddenly finds someone to lead him to the shores of safety. I helped him throughout the registration process. Since then, that student became one of my best friends. Maybe you are still thinking of our poor friend in the desert? Relax; he was lifted by a helicopter. So my job strengthens relations and builds a highly cooperative community. During my work as peer leader, I oftentimes go around talking to students, familiarizing myself with their problems and offering them the help they may stand in need of. This is just to show you an example of how a successful graduate student can positively impact the lives of other students. Finally, I recommend you to follow at least most of the five things mentioned above to be successful in your career as a graduate student.

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If you like it, please send a thank you note to Mr. Mohammed Kaabar on the AMS blog.

More later,

Nalin Pithwa

Analysis: Chapter 1: part 11: algebraic operations with real numbers: continued

(iii) Multiplication. 

When we come to multiplication, it is most convenient to confine ourselves to positive numbers (among which we may include zero) in the first instance, and to go back for a moment to the sections of positive rational numbers only which we considered in articles 4-7. We may then follow practically the same road as in the case of addition, taking (c) to be (ab) and (O) to be (AB). The argument is the same, except when we are proving that all rational numbers with at most one exception must belong to (c) or (C). This depends, as in the case of addition, on showing that we can choose a, A, b, and B so that C-c is as small as we please. Here we use the identity

C-c=AB-ab=(A-a)B+a(B-b).

Finally, we include negative numbers within the scope of our definition by agreeing that, if \alpha and \beta are positive, then

(-\alpha)\beta=-\alpha\beta, \alpha(-\beta)=-\alpha\beta, (-\alpha)(-\beta)=\alpha\beta.

(iv) Division. 

In order to define division, we begin by defining the reciprocal \frac{1}{\alpha} of a number \alpha (other than zero). Confining ourselves in the first instance to positive numbers and sections of positive rational numbers, we define the reciprocal of a positive number \alpha by means of the lower class (1/A) and the upper class (1/a). We then define the reciprocal of a negative number -\alpha by the equation 1/(-\alpha)=-(1/\alpha). Finally, we define \frac{\alpha}{\beta} by the equation

\frac{\alpha}{\beta}=\alpha \times (1/\beta).

We are then in a position to apply to all real numbers, rational or  irrational the whole of the ideas and methods of elementary algebra. Naturally, we do not propose to carry out this task in detail. It will be more profitable and more interesting to turn our attention to some special, but particularly important, classes of irrational numbers.

More later,

Nalin Pithwa

Analysis versus Computer Science

Somebody from the industry was asking me what is the use of Analysis (whether Real or Complex or Functional or Harmonic or related) in Computer Science. Being an EE major, I could not answer his  question. But, one of my contacts, Mr. Sankeerth Rao, (quite junior to me in age), with both breadth and depth of knowledge in Math, CS and EE gave me the following motivational reply:

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Analysis is very useful in Computer Science. For instance many areas of theoretical computer science use analysis – like check Pseudo randomness, Polynomial Threshold functions,…. its used everywhere.

Even hardcore discrete math uses heavy analysis – See Terence Tao’s book on Additive Combinatorics for instance. My advisor uses higher order fourier analysis to get results in theory of computer science.

Most of the theoretical results in Learning theory use analysis. All the convergence results use analysis.

At first it might appear that Computer Science only deals with discrete stuff – Nice algos and counting problems but once you go deep enough most of the latest tools use analysis. To get a feel for this have a look at the book Probabilistic Methods by Noga Alon or Additive Combinatorics by Terence Tao.

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More later,

Nalin Pithwa

On learning languages versus programming

Below are the views of the master expositor of mathematics, Paul Halmos:

Some graduate students now-a-days object to being made to learn to read two languages as a Ph.D. requirement. “Why should we learn about flowers and families and genitives and past principles? — all we want is to read last month’s Paris seminar report.” Some go further:”Who needs German? — for me Fortran (C/C++) is much more relevant.”

Horrors! I am upset and I predict that the result of such anti-linguistic, anti-cultural, anti-intellectual attitudes will lead to a deterioration of international scientific information exchange, and to a lot of bad writing. Every little bit I ever learned about any language was later of help to me as a writer. That is true of the Danish and Portuguese and Russian and Romanian that I learned for specific mathematical reasons, but it is also true of the hint or two of Greek and of Sanskrit that I managed to be exposed to. I have always rued that I was never taught Greek; every ounce of it would have paid off with a pound of linguistic insight. In the course of the years I managed to pick up quite a few Greek root words; my source of them was my shelf of English dictionaries, especially the American Heritage and the second edition of Webster. I feel that I need to look up  the etymologies of words before I can use them precisely, and I know (a small matter, but here is where it belongs) that the reason I have no trouble spelling in English is that even a nodding familiarity with other languages makes me aware of where most of the difficult words come from.

To give the devil his due, I admit that  substituting FORTRAN for German is only 90% bad, not 100. What it loses in the understanding of culture and mastering the art of communication, it gains in meticulous attention to detail and moving closer to mastering the science of communication. A knowledge of the theory and practice of formal languages might be a help for writing with precision, especially to students whose talents are not mathematical but it is of no help at all for writing with clarity. The  distinction is sometimes ignored or even argued away, but that is a sad error — there is all the difference in the world between an exposition that cannot be misunderstood and one that is in fact understood.

(From: I want to be a mathematician: An Automathography: Paul R. Halmos).

More later,

Nalin Pithwa

Chapter 1: Real Variables: examples II

Examples II.

1) Show that no rational number can have its cube equal to 2.

Proof 1.

Proof by contradiction. Let x=p/q. q \neq 0, p, q \in Z. (p, q have no common factors).

Let x^{3}=2. Hence, \frac{p^{3}}{q^{3}}=2. Hence, p^{3}=2q^{3}. Hence, p^{3} is even because we know that even times even is even and even times odd i also even and odd times odd is odd. Hence, p ought to be even. Let p=2m. Then, again q^{3}=4m^{3}. Hence, q^{3} is even. Hence, q is even. But, this means that p and q have a common factor 2 which contradicits our hypothesis. Hence, the proof.. QED.

Proof 2)

Let given rational fraction be \frac{p}{q}, q \neq 0, p, q \in Z.

Let \frac{p}{q}=\frac{m^{3}}{n^{3}}, n \neq 0, m,n \in Z.

Since p and q do not have any common factors, m and n also do not have any common factors.

Case I: p is even, q is odd so clearly, they do not have any common factors.

Case IIL p is odd, q is odd but with no common factors.

Case I: since m and n are without any common factors, and m^{3}, n^{3} are also in its lowest terms, we have p=m^{3}, q=n^{3}.

Case II: similar to case I above.

Proof 3.

A more general proposition, due to Gauss, includes those two above problems as special cases. Consider the following algebraic equation;

x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}=0.

with integral coefficients,, cannot have a rational root but non integral root.

Proof 3:

For suppose that the equation has a root a/b, where a and b are integers without a common factor, and b is positive. Writing a/b for x, and multiply both the sides of the equation b^{n-1}, e obtain

-\frac{a^{n}}{b}=p_{1}a^{n-1}+p_{2}a^{n-2}b+\ldots +p_{n}b^{n-1},

a fraction in the lowest terms equal to an integer, which is absurd. thus, b=1, and the root is a. It is clear that a must be a divisor of p_{n}

Proof 4.

Show that if p_{n}=1 and neither of

1+p_{1}+p_{2}+p_{3}+\ldots,, 1-p_{1}+p_{2}-p_{3}+\ldots is zero, then the equation cannot have a rational root.

I will put the proof later.

Problem 5.

Find the rational toots, if any of x^{4}-4x^{3}-8x^{2}+13x+10=0.’

Solution.

The roots can only be integral and so \pm 1, \pm 2, \pm 3, \pm 5 pm 10 are the only possibilities: whether these are roots can be determined by tiral. it is clear that can in this way determine the rational roots of any equation.

More later,

Nalin Pithwa

 

 

 

 

 

 

 

 

 

 

Analysis: Chapter 1: part 10: algebraic operations with real numbers

Algebraic operations with real numbers.

We now proceed to meaning of the elementary algebraic operations such as addition, as applied to real numbers in general.

(i),  Addition. In order to define the sum of two numbers \alpha and \beta, we consider the following two classes: (i) the class (c) formed by all sums c=a+b, (ii) the class (C) formed by all sums C=A+B. Clearly, c < C in all cases.

Again, there cannot be more than one rational number which does not belong either to (c) or to (C). For suppose there were two, say r and s, and let s be the greater. Then, both r and s must be greater than every c and less than every C; and so C-c cannot be less than s-r. But,

C-c=(A-a)+(B-b);

and we can choose a, b, A, B so that both A-a and B-b are as small as we like; and this plainly contradicts our hypothesis.

If every rational number belongs to (c) or to (C), the classes (c), (C) form a section of the rational numbers, that is to say, a number \gamma. If there is one which does not, we add it to (C). We have now a section or real number \gamma, which must clearly be rational, since it corresponds to the least member of (C). In any case we call \gamma the sum of \alpha and \betaand write 

\gamma=\alpha + \beta.

If both \alpha and \beta are rational, they are the least members of the upper classes (A) and (B). In this case it is clear that \alpha + \beta is the least member of (C), so that our definition agrees with our previous ideas of addition.

(ii) Subtraction.

We define \alpha - \beta by the equation \alpha-\beta=\alpha +(-\beta).

The idea of subtraction accordingly presents no fresh difficulties.

More later,

Nalin Pithwa

Chapter I: Real Variables: Rational Numbers: Examples I

Examples I.

1) If r and s are rational numbers, then r+s, r-s, rs, and r/s are rational numbers, unless in the last case s=0 (when r/s is of course meaningless).

Proof:

Part i): Given r and s are rational numbers. Let r=a/b, s=c/d, where a, b, c and d are integers, and b and d are not zero; where a and b do not have any common factors, where c and d do not have any common factors, and c and d are positive integers.

Then, r+s=a/b+c/d=(ad+bc)/bd, which is clearly rational as both the numerator and denominator are new integers (closure in addition and multiplication).

Part ii) Similar to part (i).

Part iii) By closure in multiplication.

Part iv) By definition of division in fractions, and closure in multiplication.

2) If \lambda , m, n are positive rational numbers, and m > n, then prove that \lambda(m^{2}-n^{2}), 2\lambda mn, \lambda(m^{2}+n^{2}) are positive rational numbers. Hence, show how to determine any number of right-angled triangles the lengths of all of whose sides are rational.

Proof:

This follows from problem 1 where we proved that the addition, subtraction and multiplication of rational numbers is rational.

Also, Pythagoras’ theorem holds in the following manner:

\lambda^{2}(m^{2}-n^{2})^{2}+(2\lambda m n)^{2}=\lambda^{2}(m^{2}+n^{2})^{2}

3) Any terminated decimal represents a rational number whose denominator contains no factors other than 2 or 5. Conversely, any such rational number can be expressed, and in one way only, as a terminated decimal.

Proof Part 1:

This is obvious since the divisors other than 2 or 5, namely, 3,6,7,9, and other prime numbers do not divide 1 into a terminated decimal.

Proof Part 2:

Since the process of division produces a unique quotient.

4) The positive rational numbers may be arranged in the form of a simple series as follows:

1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, \ldots

Show that p/q is the [\frac{p}{q}(p+q-1)(p+q-2)+q]th term of the series.

Proof:

Suggested idea. Try by mathematical induction.

More later,

Nalin Pithwa