i-maginary World

An interesting function June 16, 2010

This post discusses the function and another similar function of this form (I’m unsure if there is a name associated with this family of functions). The first encounter with this function is probably from Calculus on the study of indeterminate form .

First of all, we can show that is well-defined for all by taking natural logarithm,

.    (1)

Then, we can evaluate the indeterminate form by taking limit

.

Now we encounter another indeterminate form of which can be dealt with by L’Hopital’s Rule, giving

.

Hence, we have the following result

.

So far all the discussions above can be found from standard Calculus textbook. Now let us explore this function further by considering a few problems. A natural problem to look at is the derivative of . Is differentiable for ? Or in the first place, is continuous for ? The answers to both questions are the same — YES. We can of course invoke the rigorous definition but in this case it is much easier to take advantage of equation (1). Note that  for , is just a composition of continuous and differentiable functions.

It is important to emphasize that conventional differentiation rules for powers/exponents do not apply here since both the base and exponent are variable now. For instance,

.

Nonetheless, the derivative can still be computed by implicitly differentiating equation (1),

,

.

After obtaining the derivative, we can find the stationary point of by letting . Since is never 0, we have which gives . Thus, has only one stationary point . To determine whether this is a min or max point, we need to compute the second derivative

At implies that it is a minimum point. Now we can combine all the information obtained to sketch the graph of for . It will look like a skewed U-shaped curve with a min point and goes to infinity as increases.

Next, let’s turn our attention to negative values of . In particular, we would like to verify whether the following claim (posted in a math forum) is true

For is real-valued if and only if is negative integer.

One of the direction is easy to verify, i.e. if is negative integer, then is real-valued. However, the other direction is rather tricky. For example,

,

shows that can be complex-valued. Take another example says , then we’ll encounter the n-th root of unity, namely , which yields one real and two complex values. This suggests that becomes a multi-valued function and further discussion will wander far off Calculus and drift into the realm of Complex Analysis.

After experimenting with a few more examples show that is complex-valued for most of negative except at integer points. However, providing examples does not secure a mathematical proof and thus a solid proof is still sought after. My guess is we do need to use complex analysis to prove that is real-valued for negative only at integer points.

From the above discussion, we see that is rather wild at the negative side. Why don’t we deal away with the negative values by considering the following function

,

which is well-defined for all values of . At the origin, we encounter the indeterminate form again. But this time we can evaluate both one-sided limits which, after similar computations as above, give

.

Hence, in parallel to conventions, it is wise to define so that the function is now continuous and differentiable everywhere. The graph of will look like a W-shaped (with smooth corners) curve, with three turning points and symmetrical about y-axis since is an even function.

FYP May 5, 2010

It has been a very long time since I write something on this blog. The main reason is that I have been occupied with my Final Year Project (FYP) for the whole semester. As the FYP is all over now, I shall share the fruit of my hard labor here. 🙂

My project is mainly a study of a relatively new numerical methods, the moving mesh methods, for solving time-dependent PDEs. I’ll share the abstract here and if the reader is interested, please send a personal message to me for the full report.

Abstract:

Moving mesh methods have gained substantial popularity over the past two decades as an adaptive strategy to solve time-dependent partial differential equations (PDEs). The main idea of moving mesh methods is to find an invertible mapping that maps localized structure in physical domain to a smooth one in computational domain. This can be done by introducing moving mesh PDEs (MMPDEs) that constitute the core of moving mesh methods. Two important characteristics of this method are the dynamic adaptation of mesh with the solution of physical PDEs and concentration of mesh points to region of large solution variations. Various numerical experiments are performed to illustrate the idea and highlight the advantages of moving mesh methods.

Birthday primes (part 2) November 10, 2009

We have seen in the previous post that anyone whose YOB is odd will only have either 1 or 0 birthday prime. It is natural to ask what about people whose YOB is even. It turns out that from the limited observation of the table in the previous post, anyone whose YOB is even will always have at least 1 birthday prime if they live long enough. This leads me to conjecture that

Given any positive even number, we can ALWAYS find a prime such that their sum is another prime.

(Remark: I’m unsure if this is already a known result. I just find out from Wikipedia about Polignac’s conjecture which looks similar to my question. But I’m not sure if my question here is equivalent to or just a special case of the Polignac’s conjecture.)

I find no counterexample so far for the first 1 million even numbers with the first 1000 primes. I understand that 1 million is nothing compared to the infinitude of primes, but it convinces me a little that this conjecture holds. I shall sketch out my idea here:

Let be the set of all odd prime numbers and be the set of odd prime numbers strictly less than . Let be the set of all positive even numbers and for an odd prime , let . For examples,

, .

The above conjecture is then equivalent to prove that

To be continued …

Birthday primes November 8, 2009

We define birthday primes as a pair (year, age) where your age on that year are both prime numbers. I have written a Matlab program to list out all the year of birth (YOB) of people born in  the 20th century who have some positive number (#) of birthday primes (BP). The only assumption made is life span of 80 years. The YOB that are not listed have 0 birthday prime.

YOB        # of BP

1900           5
1902           6
1904           3
1905           1
1906           4
1908           6
1910           3
1911           1
1912           4
1914           6
1916           2
1918           4
1920           9
1922           3
1924           3
1926           9
1928           5
1929           1
1930           5
1931           1
1932           8
1934           3
1936           5
1938           7
1940           5
1942           4
1944           8
1946           6
1947           1
1948           3
1949           1
1950           9
1952           3
1954           3
1956          10
1958           5
1960           6
1962           6
1964           4
1966           6
1968           9
1970           7
1971           1
1972           3
1974           9
1976           6
1977           1
1978           2
1980          10
1982           6
1984           5
1985           1
1986           9
1988           5
1990           6
1991           1
1992           8
1994           5
1995           1
1996           6
1997           1
1998           7

For my own case (1986), my birthday primes are as follows:

Year          Age

1993           7
1997          11
1999          13
2003          17
2017          31
2027          41
2029          43
2039          53
2053          67

Furthermore, people whose YOB is odd can either have only 1 birthday prime or 0 birthday prime, even assuming immortality, by a simple observation:

odd YOB +2 = odd (either prime or composite)

odd YOB + odd ‘prime age’ = even (composite)

Metric function is continuous November 5, 2009

A metric on a set X is a function . For any , the function satisfies the following conditions

(i) and if and only if ,

(ii) ,

(iii) .

The purpose of this post is to prove that is a continuous function. I’m unsure whether this proposition is so trivial that it is often omitted in standard books on analysis and topology. Anyway, here’s my proof.

Let and . We define a new metric on by

We can show that this definition satisfies all the three conditions above since itself is a metric. We also require the following inequality

which can be shown by using the triangle inequality

Since we are working on metric spaces, we shall use the classical definition for a function to be continuous. Given , we set . Then for , we have

Hence, is continuous.

Door vs Set August 29, 2009

Here’s a riddle: “How is a door different from a set?”

Here’s a mathematician’s answer: “A door can be either open or closed but not both. While a set can be either open, or closed, or both, or neither!”

General Dirichlet series on Wiki August 1, 2009

As mentioned in the previous post, I have just completed an “supervised independent study” on Dirichlet series with real frequencies.

Since there is only ordinary Dirichlet series on Wikipedia, I have decided to create a new article on general Dirichlet series. This new article is like a condensed summary of my final report, with only important theorems and formulas and without any proofs.

Here is the link

http://en.wikipedia.org/wiki/General_Dirichlet_series

Enjoy ~

My 2nd URE June 18, 2009

Last year I took a undergraduate research experience (URE) project during the summer. And I take another one this summer. Well, I’ve to clarify  here that it’s more like “independent study” rather than “pure research”.

The topic of this year “project” is on Dirichlet series, which is well-studied for at least one and a half centuries. So my job is just compiling all the basic theory into one coherent report, with complete proofs to all the stated theorems.

Having googled a Harvard prof who did a genuine research project on Dirichlet series recently, I boldly send my final report to him in hope of getting some comments and advices. And he replied my email in less than 40 mins! He commented that

It’s a nice piece of work and would make marvelous lecture notes for a course.

Well, I take that as compliment because my target readers of the report are students rather than professional mathematicians as in “research papers”. I really hope that one day this report can be used as lecture notes for introductory course to Dirichlet series. Best still would be a book (with extra materials of course) to replace the good old Hardy’s book of a hundred years old. 😛

Here is the complete Final Report. Enjoy ~