254A, Supplement 3: The Gamma function and the functional equation (optional)

Originally posted on What's new:

In Notes 2, the Riemann zeta function $latex {zeta}&fg=000000$ (and more generally, the Dirichlet $latex {L}&fg=000000$-functions $latex {L(cdot,chi)}&fg=000000$) were extended meromorphically into the region $latex {{ s: hbox{Re}(s) > 0 }}&fg=000000$ in and to the right of the critical strip. This is a sufficient amount of meromorphic continuation for many applications in analytic number theory, such as establishing the prime number theorem and its variants. The zeroes of the zeta function in the critical strip $latex {{ s: 0 < hbox{Re}(s) < 1 }}&fg=000000$ are known as the non-trivial zeroes of $latex {zeta}&fg=000000$, and thanks to the truncated explicit formulae developed in Notes 2, they control the asymptotic distribution of the primes (up to small errors).

The $latex {zeta}&fg=000000$ function obeys the trivial functional equation

$latex displaystyle zeta(overline{s}) = overline{zeta(s)} (1)&fg=000000$

for all $latex {s}&fg=000000$ in its domain of definition. Indeed, as $latex {zeta(s)}&fg=000000$ is real-valued when $latex…

View original 13,726 more words

Walnut Math

A friend from China gave us a bag of walnuts plucked from their home-grown walnut tree. I decide to count them by applying math:

A stack of walnuts piled in a pyramid, with base layer arranged in a square of 6×6 walnuts, above layers 5×5, 4×4, 3×3, 2×2, and finally top 1 (1×1).

How many walnuts are there in total ? (Answer: 91)

This is simple math but only taught in A-level (with proof by induction).

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Hint: Watch free Khan Academy Math lecture to learn more ….

\displaystyle \boxed {  \sum_{1}^{n} k^2 =\frac { n (n+1)(2n+1)} {6} }

Note:
If k^2 is changed to \frac {1}{k^2} , this is Euler’s “Basel Problem” which leads to the unsolved Riemann Hypothesis.

Below is a 400-year-old walnut tree: walnut is called “Wise fruit 聪明果”, it looks like human brain, also has proven nutritious benefits to brain.

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Stop Teaching Calculating, Start Learning Maths!

The British Wolfram company is the maker of the Math Computer Tool “Methematica” used for university students to prove theorems besides computing. It is written in Lisp functional language.

Conrad Wolfram provoked the new idea of Computer-Based Math education:

Teach the ‘Why’ of Maths, leave the ‘How’ to the computer.

‘How’: solve quadratic equation, simultaneous equations, differentiation, integration….

He mentioned Singapore is interested in this new approach of teaching Math ? The O & A Level students can now use scientific calculator in Exams.

Conrad Wolfram comes from the British angle, telling us the English Math is too computational, or Applied. It is like a fast-food chef without knowing much about the food science, the temperature and the art of color 色, taste 香, smell 味. The French Math is, au contraire, theoretical, like the ‘haute cuisine’ chef, more “Why” than “How”, slowly cook and taste the finest quality of food (dégustation 品尝).

Having taken GCE A-level Math and French Classe Prepas Math, I experienced the strengths and weaknesses of both Maths. Giving an Integration question to an English educated student, he would immediately plunge into all sorts of techniques (by substitution, by parts…) and get the answer; give it to a French student, he would first study the ‘domain of definition’ of the function, continuity at which intervals, … before attempting to integrate if it is “integritable” — meaningful to integrate or has any solution — in the first place !!

A best Math education is the combination of both English and French. The computer can be used to calculate faster in the (English) Applied Math, to verify / prove the (French ) Theoretical Math.

It is like the 少林巭 Shaolin Kungfu (English Applied Math ) versus 武当太极 Wudang Taiji (French Pure Math). You can have both !

Stephen Wolfram: Computing a theory of everything

Stephen Wolfram: Founder & CEO of Mathematica (UK)

Wolfram Alpha: Knowledge-base Computing using public data on the net and private information.

Example: Calculate : For Year 2014
Spain GDP / Google Revenue

Mathematica: Math tool using Symbolic Functional Language (LISP)

New Kind of Science: Cell Automata

Physics: From Computing World to find new Physical World

Alibaba Arithmetic

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Jack Ma of the Alibaba.com gives this Arithmetic question to the audience,  only 1% get the answer right !

Jack has cash $50, which he uses to buy:
Clothing : $20 (Balance $50-20 = $30)
Shoes: $15 (Balance = $30-15 = $15)
Candy: $9 (Balance = $15- 9 = $6)
Food: $6 (Balance = $6 – 6=$0)

Question:
Add up the Balances = $ 30+15+6 = $51

Where does the extra $1 come from ?

No excuse for Accountants to get it wrong !!!☺

[Answer: Scroll down below at last page….]

On a more serious Accounting topic. There was a Management Conference orgainsed by the Chairman of a big MNC, during the first day he asked the Human Resource Director to emphasise that employees are valuable Assets of the company, and the second day the Finance Director on the importance of keeping a healthy Balance Sheet.

A Sales Manager, fed-up of the recent massive employees lay-off, raised a question to the Chairman on the 3rd day during the Q&A session:

“Mr. Chairman, we learn from HR that employees are valuable Assets, why aren’t they appeared on the Finance Balance Sheet ?”

There was an uproar clapping of hands and laughters in the hall.

Both HR and Finance directors were embarrassed by the question. The enraged Chairman took the microphone,

“Very simple answer! To me, those employees who contribute to the bottom-lines are Assets, those who don’t are the Liabilities!”

All the managers kept silence, in their mind they started to draft their C.V.

Answer of Alibaba Arithmetic: (If you don’t know the answer, please revise your Accounting….Scroll down below)
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In Accounting rule: Debit = Credit
Credit Cash = $50
Debit Expenses = $ 20 + 15 + 9 +6 =$50
No meaning to add Balances. If my first 2 transactions were $1, $1, the Balance would be $49, then $48, add them up = $ 49 + $48= $97 > $50. Absurd!