Posts Tagged ‘discovery’

Did Albert Einstein retire?

November 26, 2013

Have you done reading about Albert Einstein?

I wonder if, after retiring, he gave up seeking to understand the mysteries of the universe?

I find it hard to believe that he would. From what I have read, he did it for the pure love and curiosity, not because it was “his job.”

I’ll bet he was thinking about the mysteries of the universe up till his last day.

Do you know?

Update (November 27, 2013):

A friend sent me this note:

There is the yellow legal pad he used the
day he died. I saw it in a museum somewhere.
It was scribbled with equations from his work
on unifying the four forces.

Awesome!

Learning for the sheer love of it … determining the number of edges in a complete graph

September 7, 2013

As I noted in an earlier blog, even little discoveries makes one feel wonderful. Today I solved a small problem (i.e., made a little discovery) and I feel wonderful.

I am reading a wonderful book, Introduction to Graph Theory. I am at the part where it describes complete graphs. A complete graph is one where each vertex is connected to every other vertex. The author asks, How many edges are in a complete graph with N vertices? I took this as a challenge to use my reasoning skills to figure it out.

Here are three examples of complete graphs and the number of edges in each graph:

Three examples of complete graphs and the number of edges in each graph

Let’s arbitrarily choose one vertex. It must connect to the other (n-1) vertices. For example, in the complete graph with 4 vertices, the vertex will connect to the other 3 vertices:

A vertex connects to n-1 other vertices

The remaining vertices must form a complete graph amongst themselves. Ah Ha! Now we have a simpler problem: find the number of edges in a complete graph of n-1 vertices.

I now switch on my programmer hat. I smell recursion (which I love):

numEdges (N) {
    if (N == 2) then return 1
    else return (N-1) + numEdges (N-1)

Let’s check that program:

if N = 2, then num edges = 2
if N = 3, then num edges = 2 + 1 = 3
if N = 4, then num edges = 3 + 3 = 6
if N = 5, then num edges = 4 + 6 = 10
if N = 6, then num edges = 5 + 10 = 15
if N = 7, then num edges = 6 + 15 = 21
if N = 8, then num edges = 7 + 21 = 28

Wow! That is cool.

One problem, however. What are the number of edges when, say, N = 1000?

I could implement the above program and set it to solve N=1000. But I know the mathematicians would tell me: Find a formula.

Okay, let’s see … to find the number of edges in a complete graph with N vertices, I must add these numbers:

(N-1) + (N-2) + (N-3) + … + 1

or equivalently:

1 + 2 + … + (N-1)

I remember from school days a formula for the sum of the first N integers:

n(n+1)/2

Of course, I want the sum of the first N-1 integers, so I substitute “n” with N-1:

(N-1)((N-1)+1)/2
= (N-1)(N-1+1)/2
= (N-1)(N)/2
= N(N-1)/2

Let’s check:

if N = 2, then 2(1)/2 = 1
if N = 3, then num edges = 3(2)/2 = 6
if N = 4, then num edges = 4(3)/2 = 6
if N = 5, then num edges = 5(4)/2 = 10
if N = 6, then num edges = 6(5)/2 = 15
if N = 7, then num edges = 7(6)/2 = 21
if N = 8, then num edges = 8(7)/2 = 28

Yea!

I am pretty proud of myself. I solved it, not because it was a homework problem or a work problem. I solved it purely for the love of learning. I figured it out by simple reasoning.

I feel wonderful.

There is discovery in solving every problem

September 2, 2013

A great discovery solves a great problem,
but there is a grain of discovery in the solution
of every problem. Your problem may be modest,
but if it challenges your curiosity and brings into
play your inventive facilities, and if you solve it
by your own means, you may experience the
tension and enjoy the triumph of discovery.

George PĆ³lya, How To Solve It

Our society is moving toward short-term thinking, quick profits

July 21, 2013

The latest issue of Time magazine has a fascinating article about The Institute for Advanced Study in Princeton, N.J. Recall that this is where Albert Einstein did his work on relativity.

The institute offers researchers a place to work unhindered by the pesky objectives required by traditional research centers or obligations to students at universities. If research were measured on a spectrum from the practical (like making a laptop slimmer) to the theoretical (like studying the way matter moves in space), the institute is as “close to the frontier as possible,” says its new director, Dutch mathematical physicist Robert Dijkgraaf. The atmosphere, free from practical constraints and flush with great minds — 33 Nobel laureates have stopped through along with more than two-thirds of the winners of the Fields Medal, math’s top honor since 1936 — was designed to create ideal conditions for discovery. And in many cases over the years, it has. In its prolific early history, physicists Chen Ning Yang and Tsung-Dao Lee discovered that nature is not symmetrical, mathematician John von Neumann created the prototype on which future computers were built, and historian and diplomat George Kennan developed the intellectual foundations of realist foreign policy.

But while the institute’s future is mostly assured by a healthy $650 million endowment, elsewhere in the U.S. the kind of work it fosters is staring down the barrel of a rough future as the government spends less than in past decades on research as a share of GDP. Many countries in Europe and Asia, meanwhile, are heading the other direction. The Chinese government — coming from far behind — has been pouring money into research and development at such a clip that according to one estimate from Battelle, a non-profit technology-development group, its spending will surpass America’s in just 10 years. Chinese universities, which hardly awarded doctorates years ago, are now, albeit barely, competing with the U.S. for students. The U.S. is already falling in some measures of innovation: in the decade from 2000 to 2010, the U.S. share of science and engineering academic citations dropped precipitously, putting it in close competition with the E.U.

If the institute was once a symbol of all that was powerful about American innovation, now it is emblematic of some of the toughest questions facing the U.S. in its fight to stay competitive. Demands for quick results are everywhere, from corporations focused on quarterly reports to universities increasingly obsessed with private-enterprise partnerships that can spawn start-ups and burnish their image with students and donors. “I feel the institute is a little bit the canary in the mine,” says Dijkgraaf, 53, who took his post last summer. “It is not clear a place like this can exist. Society is moving toward short-term thinking, toward direct applications. We are swimming against the stream.” In other words, pursuing questions for which the value of the answers isn’t obvious may be a luxury that America can no longer afford — or at least appreciate the importance of.