The WordPress.com stats helper monkeys prepared a 2015 annual report for this blog.
Here’s an excerpt:
The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 27,000 times in 2015. If it were a concert at Sydney Opera House, it would take about 10 sold-out performances for that many people to see it.
You need to take SAT during JC2 year to apply for USA universities, a good SAT score is more important than GCE A-level.
John borrows $500 from dad and $500 from mom. He uses $970 to buy a pair of shoes, balance $30.
He returns $10 to dad, $10 to mom, keeps $10 for himself.
So he owes $490 to dad, $490 to mom,
$490 + $490 = $980,
with his $10 = $990.
Where is the missing $10 (= $500 + $500 – $990) ?
This 3rd Isomorphism Theorem can be intuitively understood as:
G partitioned by a bigger normal subgroup H
is isomorphic to:
{G partitioned by a smaller normal subgroup K (which is a subgroup of H)}
partitioned by
{H partitioned by a smaller normal subgroup K}
or, by ‘abuse of arithmetic’: divide G & H by a common factor K.
Analogy:
$100 / $50 = 2 (two $50 notes makes $100)
is same (isomorphic) as
$100 / $10 = 10, (ten $10 notes makes $100)
$50/$10 = 5, (five $10 notes makes $50)
then 10/5 = 2 (ten notes split into five is two )
I found this “lattice diagram” only in an old Chinese Abstract Algebra Textbook, never seen before in any American/UK or in French textbooks . Share here with the students who would find difficulty remembering the 3 useful Isomorphism Theorems.
Reference: 2nd Isomorphism Theorem (“Diamond Theorem”)
Let G be a group. Let H be a subgroup of G, and let N be a normal subgroup of G. Then:
1. The product HN is a subgroup of G,
The intersection H ∩ N is a normal subgroup of H, and
2. The 2 quotient groups
(HN) / N and
H / (H∩ N)
are isomorphic.
It is easy to remember using the green diagram below: (similarly can be drawn for 1st & 3rd Isomorphism)
This 2nd isomorphism theorem has been called the “diamond theorem” due to the shape of the resulting subgroup lattice with HN at the top, H∩ N at the bottom and with N and H to the sides. It has even been called the “parallelogram rule” (by analogy with the parallelogram rule for vectors) because in the resulting subgroup lattice the two sides assumed to represent the quotient groups (HN) / N and H / (H ∩ N) are “equal” in the sense of isomorphism.
Note: Technically, it is not necessary for N to be a normal subgroup, as long as H is a subgroup of the normalizer of N. In this case, the intersection H ∩ N is not a normal subgroup of G, but it is still a normal subgroup of H.
Note: This theorem still applies to Ring, just replace normal subgroup by ideal.
石墨烯将取代硅 (Silicon)，为世界电子科技开创一个崭新的时代！
石墨烯手机充电时间只需5秒，电池就满档，可以连续使用半个月！
石墨烯电池只需充电10分钟，环保节能汽车就有可能行驶1000公里！
A Chinese Mathematician Figured Out How To Always Win At Rock-Paper-Scissors – (Business Insider)
This is “Game Theory” demonstrating the Nash Equilibrium.
Very good to understand the “Kia-Soo” (Singlish means: 惊(怕)输 “afraid to lose”) syndrome of Singaporeans.
To win this game and beat the “kia-soo” mentality — 反其道而行 Adopt the reverse way of the opposition’s anticipated kia-soo way 🙂
Key points:
(1). Sequence : “R- P -S” or (中文习惯) “石头 – 剪刀 – 布”;
(2). Winner tends to stay same way in next move;
(3). Loser likely to switch to the next step in the Sequence (1).
Reflection:
In business,
(2) is where big conglomerates like IBM , HP, Sony, Microsoft etc lose because they stay put with the same strategy (Corporate Data Center, Sell thru Channel distributors with mark-up, CD/DVD music… ), and products (Mainframes, Servers, PC, CRT-TV, Packaged software…) which brought them to success but never re-invent themselves.
[A sad case is Kodak Photo]
(3) is the new start-ups like Google, Facebook, Amazon, Alibaba, Apple (after Steve Jobs rejoined in 1997), etc. They change the traditional way of doing businesses in (2) by using “Disrupted Technologies” (Search, Cloud, Smartphone, Tablet, Social Networking, …) and processes (AppStore, eBook, eCommerce, …).
http://www.businessinsider.sg/rock-paper-scissors-strategy-2014-6/#.VmepoEncjkR.whatsapp
Note:
Nash was the American Mathematician in the movie “Beautiful Mind”. He won Nobel Prize in Economics for the Game Theory – “Nash Equilibrium”.
The sum of all positive integers is divergent, but the result is ( -1/12).
Crazy but it is true in Quantum Physics (String Theory) !
2nd Proof:
Math Online Tom Circle 