A very meaningful sentence from Introduction to Proofs and Real Analysis, written by Richard C. Penney:

“There is a fundamental diﬀerence between mathematics and other sciences. In most sciences, one does experiments to determine laws. A “law” will remain a law, only so long as it is not contradicted by experimental evidence. Newtonian physics was accepted as valid until it was contradicted by experiment, resulting in the discovery of the theory of relativity.

Mathematics, on the other hand, is based on absolute certainty. A mathematician may feel that some mathematical law is true on the basis of, say, a thousand experiments. He/she will not accept it as true, however, until it is absolutely certain that it can never fail. Achieving this kind of certainty requires constructing a logical argument showing the law’s validity–i.e. constructing a proof.”

I was reading some PhD guidelines and happened to come across a very meaningful remark. I felt exciting and decided to copy the inspiring paragraph here.

Don’t teach!

. . . more than you have to. For many, teaching is attached to astipend or is otherwise economically unavoidable. In this case, dowhat you must! Moreover, there are some real intellectual andpractical advantages from doing a couple of terms of TA work. Explaining the concepts to others is very useful in consolidating themin yourself. But beyond this, the returns become strongly negative. Your job is research – and anything that distracts you from this is aheavy cost. The first cost, which may seem remote at the time that youare deciding on the teaching, is that it could delay completion of thethesis by a year or more. An even larger cost is if it crowds out timeto write a really great thesis. As a PhD student, your time is veryvaluable; treat it that way.

Well. Some people say yes. Some people will say no and explain that the radical number 0.999… is approaching 1 but is not 1. The value should be somewhere near to 1 but is not exactly 1.

So, is 0.999… = 1?

I not sure but there is a simple proof that supports the claim.

Let (1)

Multiply (1) by 10,
(2)

Then (2)-(1):

From Mathematical Mysteries – The Beauty and Magic of Numbers by Calvin C. Clawson.

Someone ask me to explain the different between random variables and random process. Here are my explanation.

Say we are a wholesaler of apples in a city. One day a worker moves down a bucket of apples from a truck. By looking at the apples in this bucket, we can measure the expected weight and variation of apples in this bucket. Let denote the expected weight as , in which it is a random variable that possesses finite mean and variation.

Then, the worker moves down another bucket of apples. We not sure if this bucket of apples possess the same statistical properties as the previous bucket. By intuition, we know these apples in the second bucket possesses a finite expected weight as well (or you will see a HUGE apple!!). Next, the worker continue to move down a lot of buckets of apples from the truck (big truck!). For convenience, we denote the expected weight as where denotes the sequence where the worker moves down the buckets.

The manager of the wholesaler is not interested at the weight distribution of apples in any particular bucket but among certain groups of buckets. For example, he wants to know if the first bucket has worm, what is the chance that the nth bucket has worm as well. This question the inter-relation among the buckets. So, we have no choice but to extend the notion “a single bucket” into “groups of buckets”. In mathematics, we write this as and call this the random process (or stochastic process) of X. In such a way, we can describe the correlation (or other statistical properties) between the first bucket and the second bucket or any bucket in the later sequence to the manager.

As for the summary, in random process (stochastic process) we are dealing with a group of “homogeneous” random variables in the sense that they are of the same function (e.g. apples) but they may possess different distribution function.