Negative

Binomial Theorom

electron — Wed, 31 Oct 2007 06:01:56 +0000

– (i)

Proof:

When n = 1

Assume that (i) is true for n = m.

Observe that:
– (ii)

– (iii)

– (iv)

From (ii), (iii) & (iv)

QED

Principle of finite Induction.

electron — Tue, 30 Oct 2007 10:33:30 +0000

Let S be a set of positve integers with the properties

1 belongs to S, and
whenever the integer k is in S, then the next integer k + 1 must also be in S.

Then S is the set of all positive integers.

Proof: Let T be the set of all positive integers not in S, and assume that T is nonempty. The Well-Ordering Principle tells us that T possesses a least element, which we denote by a. Since 1 is in S, certainly a > 1 so 0 < a – 1 < a. The choice of a as the smallest positive integer in T implies that a – 1 is not a member of T, or equivalently, that a – 1 belongs to S. By hypothesis, S must also contain (a – 1) + 1 = a, which contradicts the fact a lies in T. We conclude that the set T is empty, and in consequence that S contains all the positive integers.

Archimedean Property

electron — Tue, 30 Oct 2007 10:12:24 +0000

If a and b are any positive intergers, then there exists a positive integer n such that

Proof:

Assume that the statment of the theorem is not true, so that for some a and b, for every positive integer n. Then the set

consists entirely of positive integers. By the Well-Ordering Principle S will posses a least element, say . Notice that also lies in S, since S contains all integers of this form. Furthemore, we have

contrary to the choice of as the smalled integer in S. This contradiction arose out of our original assumption that the Archimedian property did not hold, hence this property is proven true.

Well Ordering Principle

electron — Tue, 30 Oct 2007 09:41:25 +0000

Every nonempty set S of nonnegative integers contains a least element; that is; there is some integer a in S such that for all b belonging to S.

Kids, kids, me, kids, kids …

electron — Fri, 02 Mar 2007 13:44:10 +0000

Group Photo

Sweet friends.

I want to be in the photograph.

(courtesy Suman)

Had a fun day yesterday, outing to Cubbon park to manage (play with) parikrama kids. Played with kids, lots of them. They were tiny and energetic (like electrons ). Check them out at my flickr stream. I got a marble from a kid.

12 Step NAS

electron — Thu, 01 Mar 2007 02:02:22 +0000

I am a member of this group called Nikon-D50. NAS == Nikcon Acquisition Syndrome. Here is what a member named Scott Gibson writes about how to get NAS.

12 step NAS program.

Step 1 50mm F1.4 or f1.8
Step 2 18-200 vr
Step 3 SB 600 or SB 800
Step 4 second VR lens
Step 5 first wide angle lens
Step 6 SB 800
Step 7 second wide angle lens
Step 8 105mm Micro
step 9 Upgrade Camera Body
step 10 R1C1 lighting kit
step 11 Long prime lens
step 12 Second Micro or long prime lens

Gentoo Installation

electron — Sun, 26 Feb 2006 02:42:09 +0000

God it was most tiring experience in installing linux for me. I started @ 2 AM in the morning. I went to gentoo.org started reading the manuals. At the same time downloading stage3 tarball and the portage snapshot. At the end of it I had a huge notes on what to do!! The installation was with gentoo minimal cd, so net connection is a must. Infamous murphy kicked in. I was not able to connect to the net even after configuring the ehernet adaptors. I struggled with it for 1/2 hour before realising that for some reason my network cards were detected in reverse order. I was configuring it like I usually configure on my linux. So I disabled eth0 and configured eth1, net connection up. Cool, now I did mke2fs crossing my fingers on my old linux system. Its gone for good. unrolled the tarballs, and did an emerge -sync. It took an hour to finish. And then the kernel installation another hour for configuring and installing it. Installed grub and then tried to install it on the bootsector it started cribbing the my /boot was not there or is not a block device. Said get lost and used the grub shell to install the bootloader. Thanks to the default installation there is no vi/vim here only nano. I keep typing all vim commands here. I messed up my /etc/fstab. gentoo refused to boot. For my luck I messed up the first line which is the / config line. Hell with it, I had to boot back with gentoo CD and then edit the conf file and boot in. Cool, I get a prompt. Ok now what is my password. The note said something like it is scrambled password. Why cant my life be simple. I had to boot back with Gentoo CD and set the root password then mez back. Cool now it boots in and can login as root. No net connection ???. Damn I configured eth1, now its reverted to eth0. Not so bad, reconfigure and start eth0. Net it up. Its 7 in the morning already. my sleep!!! OK I have a base system, no vim, no X nothing on it. I installed enlightenment. startx ouch no xorg.conf. To xorgcfg. OMG no mouse detected, curse you gentoo. What is the mouse device? /dev/mouse not there ( I felt like killing it (kill -9 gentoo). Ah there you see there is /dev/input directory. Check it there is mouse0 cool. Now edit xorg.conf with the new device. /dev/input/mouse0 X comes up what 25% of my screen in down. Set it right now enlighenment. What is available? XTerm thats it. My system stands now with just enlighenment, xterm, python, gvim (gtk12), no sound, nothing else configured. Mez tired. 7 hours of installation and get bare bones. Price I pay for configuring all myself. Still a cool experience. Hope I get my gentoo in full pace in a week.

HackThisSite.org

electron — Tue, 01 Mar 2005 15:32:57 +0000

Today, felt a lot bored. Did not have much to do. My hand broken and cant go to office and all. As always I was browsing the del.icio.us page. Found a cool site named http://www.hackthissite.org/. Registered and started the basic test. Initially I thought it was a tutorial. But I soon realised that it was a test and got more interested. Cleared 8 levels. The encryption thingy took sometime. For my bad luck the net connection went down. Shud try the rest tommorow. It is very interersting. I get to learn a lot when I try to break the puzzles. Some were damn easy. Look at it you have the solution. Some take quite some time. Do not know how many levels are there. Do not know how long it will take. Currently stuck at the 9th level.

Apart from this nothing interesting today. I discovered MSN Spaces and put my first blog entry.
Hmmmm… shud work more on the hacking part. Quite interesting.