Chain Rule of Differentiation

Consider the curve y = f(x).

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\displaystyle{\frac{d}{dx}} is an operator, meaning “the slope of the tangent of”. So the expression \displaystyle{\frac{dy}{dx}}, meaning \displaystyle{\frac{d}{dx} (y)}, is not a fraction.

In order words, it means the slope of the tangent of the curve y = f(x) at a point, such as point A in the graph.

d_2018_07_15__21_31_32_PM_

The symbol dx has no relation with the symbol \displaystyle{\frac{dy}{dx}}. It means \Delta x as shown in the graph. In other words,

dx = \Delta x

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The symbol dy also has no relation with the symbol \displaystyle{\frac{dy}{dx}}. It means the vertical distance between the current point A(x_0, y_0), where y_0 = f(x_0), and the point C on the tangent line y = mx + c, where m is the slope of the tangent line. In other words,

dy = m~dx

or

\displaystyle{dy = \left[ \left( \frac{d}{dx} \right) y \right] dx}

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The relationship of \Delta y and dy is that

\displaystyle{\Delta y = \frac{dy}{dx} \Delta x + \text{higher order terms}}

\displaystyle{\Delta y = \frac{dy}{dx} dx + \text{higher order terms}}

\Delta y = dy + \text{higher order terms}

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Similarly, for functions of 2 variables:

\displaystyle{\Delta f(x,y) = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + \text{higher order terms}}

\displaystyle{df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy}

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For functions of 3 variables:

\displaystyle{df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz}

\displaystyle{\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} + \frac{\partial f}{\partial z}\frac{dz}{dt}}

— Me@2018-07-15 09:30:29 PM

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2018.07.15 Sunday (c) All rights reserved by ACHK

Problem 14.4b2

Closed string degeneracies | A First Course in String Theory

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(b) State the values of \alpha' M^2 and give the separate degeneracies of bosons and fermions for the first five mass levels of the type IIA closed superstrings. Would the answer have the different for type IIB?

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Type IIB closed superstrings

Equation (14.85)

(NS+, NS+), (NS+, R-), (R-, NS+), (R-, R-)

— Me@2015.09.16 06:08 AM: Should be the same. But I am not sure whether I have missed something.

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f_{NS+}(x) = 8 + 128 \, x + 1152 \, x^{2} + 7680 \, x^{3} + 42112 \, x^{4} + ...

f_{R-}(x) = 8 + 128 x + 1152 x^{2} + 7680 x^{3} + 42112 x^{4} + ...

f_{NS-}(x) = \frac{1}{\sqrt{x}} + 36 \sqrt{x} + 402 x^{\frac{3}{2}} + 3064 x^{\frac{5}{2}} + ...

f_{R+}(x) = 8 + 128 x + 1152 x^{2} + 7680 x^{3} + 42112 x^{4} + ...

— Me@2018-07-14 09:41:10 PM

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2018.07.14 Saturday (c) All rights reserved by ACHK

Pointer state

Eigenstates 3

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In quantum Darwinism and similar theories, pointer states are quantum states that are less perturbed by decoherence than other states, and are the quantum equivalents of the classical states of the system after decoherence has occurred through interaction with the environment.

— Wikipedia on Pointer state

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In calculation, if a quantum state is in a superposition, that superposition is a superposition of eigenstates.

However, real superposition does not just includes states that make macroscopic senses.

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That is the major mistake of the many-worlds interpretation of quantum mechanics.

— Me@2017-12-30 10:24 AM

— Me@2018-07-03 07:24 PM

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2018.07.03 Tuesday (c) All rights reserved by ACHK

Mirror selves, 5.2

Anatta 3.3 | 無我 3.3

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You fight for existence, for being alive.

However, your existence is not “yours”.

The existence of you, is not your property.

The existence of you, is a property of the group you are in.

The existence of you, is a property of other people.

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To meaningfully say the statement “I exist”, you have to specify you exist with respect to whom.

To exist, you have to specify to exist in which people’s world.

— Me@2018-05-22 7:43 AM

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2018.07.02 Monday (c) All rights reserved by ACHK

神的旨意 2.1

這段改編自 2010 年 4 月 18 日的對話。

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(問:你閱讀過很多有關「瀕死經驗」的文章?)

可以這樣說。

如果你閱讀那些文章的話,要小心一點,因為那類文章良莠不齊——當中有些文章發人深省,有些則謊話連篇。

(問:那你怎樣分辨,「瀕死經驗」的文章之中,哪些是真,哪些為假?)

看看文中所說的,合不合理。

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例如,你怎樣知道,我說的話,是真還是假?

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合情合理,自圓其說的,就有機會真;無情無理,自相矛盾的,則必定為假。

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又例如,如果現在神明顯靈在你面前,你怎樣判斷,那真的是「神明」?

「祂」既可能其實是「邪靈」,亦可能只是你自己的幻覺而已。

— Me@2018-06-28 10:23:28 PM

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2018.06.30 Saturday (c) All rights reserved by ACHK

Block spacetime, 9

motohagiography 42 days ago [-]

I once saw a fridge magnet that said “time is natures way of making sure everything doesn’t happen all at once,” and it’s stuck with me.

The concept of time not being “real,” can be useful as an exercise for modelling problems where to fully explore the problem space, you need to decouple your solutions from needing them to occur in an order or sequence.

From an engineering perspective, “removing” time means you can model problems abstractly by stepping back from a problem and asking, what are all possible states of the mechanism, then which ones are we implementing, and finally, in what order. This is different from the relatively stochastic approach most people take of “given X, what is the necessary next step to get to desired endstate.”

More simply, as a tool, time helps us apprehend the states of a system by reducing the scope of our perception of them to sets of serial, ordered phenomena.

Whether it is “real,” or an artifact of our perception is sort of immaterial when you can choose to reason about things with it, or without it. A friend once joked that math is what you get when you remove time from physics.

I look forward to the author’s new book.

— Gödel and the unreality of time

— Hacker News

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2018.06.26 Tuesday ACHK

Quick Calculation 14.8.2

A First Course in String Theory

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What sector(s) can be combined with a left-moving NS- to form a consistent closed string sector?

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There are no mass levels in NS+, R+, or R- that can match those in NS-. So NS- can be paired only with NS-:

(NS-, NS-)

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f_{NS} (x)
= \frac{1}{\sqrt{x}} \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8
= \frac{1}{\sqrt{x}} g_{NS}(x)
= \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...

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g (\sqrt{x})
= \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8
= 1 + 8 \, \sqrt{x} + 36 \, x + 128 \, x^{\frac{3}{2}} + 402 \, x^{2} + 1152 \, x^{\frac{5}{2}} + 3064 \, x^{3} + ...

g (-\sqrt{x})
= \prod_{n=1}^\infty \left( \frac{1-x^{n-\frac{1}{2}}}{1-x^n} \right)^8
= 1 -8 \, \sqrt{x} + 36 \, x -128 \, x^{\frac{3}{2}} + 402 \, x^{2} -1152 \, x^{\frac{5}{2}} + 3064 \, x^{3} + ...

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g (\sqrt{x}) + g (-\sqrt{x})
= 2(1 + 36 x + 402 x^{2} + 3064 x^{3} + ...)

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f_{NS-}(x)
= \frac{1}{2 \sqrt{x}} \left[ g (\sqrt{x}) + g (-\sqrt{x}) \right]
= \frac{1}{2 \sqrt{x}} \left[ \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8 + \prod_{n=1}^\infty \left( \frac{1-x^{n-\frac{1}{2}}}{1-x^n} \right)^8 \right]
= \frac{1}{2 \sqrt{x}} \left[ 2(1 + 36 \, x + 402 \, x^{2} + 3064 \, x^{3} + ...) \right]
= \frac{1}{\sqrt{x}} + 36 \sqrt{x} + 402 x^{\frac{3}{2}} + 3064 x^{\frac{5}{2}} + ...

— Me@2018-06-26 07:36:41 PM

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2018.06.26 Tuesday (c) All rights reserved by ACHK

Eigenstates 2.3.2

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eigenstates

~ classical states

~ definite states

— Me@2012-04-15 11:42:10 PM

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The concept of eigenstate is relative.

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First, you have to specify the eigenstate is of which physical observable.

A physical system can be at an eigenstate of one observable but at a superposition state of another observable.

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Second, you have to specify the state of that observable is eigen with respect to which observer.

— Me@2018-06-16 7:27 AM

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eigenstates

~ of which observable?

~ with respect to which observer?

— Me@2018-06-19 10:54:54 AM

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2018.06.19 Tuesday (c) All rights reserved by ACHK

大學經濟

這段改編自 2010 年 4 月 18 日的對話。

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我猜想,當一個人改變存在型態時,會立刻或者將會,知道很多生時不知道的東西。但是,那些新知識,未必包括你想知道的東西。

比喻說,由中學升到大學,你將會學到,很多超過中學程度的知識。但是,如果你中學時,沒有讀過經濟科的話,單單是「升大學」本身,並不會令你,立刻獲得經濟科的知識。

大學生「由零開始學經濟學」,都同樣要花時間;分別是,通常而言,比中學生「由零開始學經濟學」,速度會高一點。

(問:不一定呀。中學生比較年青,腦袋理應高速一點。)

無錯。

方便起見,暫時用同一個人來比較,例如你。

「中學的你」可以因為腦袋較年青,學習新事物比「大學的你」較快。「大學的你」可能因為知識和經驗較多,學習新事物比「中學的你」較快。

視乎情況,因人而異,沒有一定的答案。

但是,至少你會同意一點:

如果你中學時,沒有讀過經濟科,在大學時要「由零開始學經濟學」的話,你會立刻看大學程度的經濟書,而不是由中學教科書開始學。

— Me@2018-06-05 11:54:51 AM

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2018.06.07 Thursday (c) All rights reserved by ACHK

Plato

trowawee 4 months ago

I’m a little frustrated at the tossed-off reference to Plato and Aristotle at the beginning – “The good life may have sufficed for Plato and Aristotle, but it is no longer enough.” – because I feel like that ignores the fact that both Plato and Aristotle, along with a lot of philosophers, actually had a lot to say about physical fitness. Plato was a champion wrestler, and both he and Aristotle viewed physical education as a fundamental component to living the good life. Xenophon quotes Socrates saying this:

“For in everything that men do the body is useful; and in all uses of the body it is of great importance to be in as high a state of physical efficiency as possible. Why, even in the process of thinking, in which the use of the body seems to be reduced to a minimum, it is matter of common knowledge that grave mistakes may often be traced to bad health.”

The whole article feels a little too mired in presentism, and ignorant of the history of self-improvement ideas.

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coldtea 4 months ago

>Plato was a champion wrestler

And the name Plato is a nickname — meaning “the broad/wide one” given to him for his broad shoulders because of that training and physical appearance. Real name: Aristocles.

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kernelbandwidth 4 months ago

It’s funny to consider that one of the canonically great philosophers in history is known essentially by the equivalent of his WWE wrestling name. It’s like if in the future there were classes taught on the philosophical ideas of “The Rock”.

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coldtea 4 months ago

Some other amusing related stuff: so, Plato, was called for for the ancient greek word for broad/wide.

Modern [English] words that stem from the same root: plateau, platitude, plat, plate — via French and Latin (plattus) from Greek (platis “flat, wide, broad”).

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danohu 4 months ago

Well, the first Pope was literally called The Rock (Peter). Jesus appointed him by saying “you are The Rock, and I’ll build my church on this rock”.

Exactly what he meant has led to centuries of debate between protestants and catholics.

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acangiano 4 months ago

“No man has the right to be an amateur in the matter of physical training. It is a shame for a man to grow old without seeing the beauty and strength of which his body is capable.”

― Socrates

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— Improving Ourselves to Death

— Hacker News

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2018.06.01 Friday ACHK

Problem 14.4b1.4

Closed string degeneracies | A First Course in String Theory

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What is the meaning of “With a = 1, ..., 8 and \bar b = \bar 1, ..., \bar 8, …”?

— Me@2015.09.14 12:11 PM

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p.315 “Explicitly, the eight states | R_a \rangle, a = 1, 2, ..., 8, with an even number of creation operators are … ”

p.316 “The eight states |R_{\bar{a}} \rangle, \bar a = \bar 1, \bar 2, ..., \bar 8, with an odd number of creation operators are … ”

— Me@2018-05-24 11:41:34 AM

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2018.05.24 Thursday (c) All rights reserved by ACHK

Life, 3

生命 3

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We exist in time because time is change.

Growing is part of the definition of life. Growing is a kind of change.

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Also, without time/change, there would be no thinking and no thoughts.

— Me@2017-12-26 11:42 am

— Me@2018-05-23 10:05:03 PM

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time ~ change

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Time is logically necessary if change is necessary.

— Me@2018-02-04 09:07:48 PM

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2018.05.23 Wednesday (c) All rights reserved by ACHK

多項選擇題 6

Multiple Choices 6

這段改編自 2010 年 8 月 24 日的對話。

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有時,一題物理 MC(多項選擇題)會,同時有數學做法和物理做法。

那時,你就先用物理方法做一次,再用數學方法做多一次,以作驗算。

(問:哪有那麼多的時間?)

之前講過,那些做法,必須透過考試前,平日多加收集和練習而來;並不是在考試中途,才花額外時間發明。

— Me@2018-05-22 06:02:40 PM

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2018.05.22 Tuesday (c) All rights reserved by ACHK

Quantum Computing, 2

stcredzero 3 months ago

A note for the savvy: A quantum computer is not a magic bit-string that mysteriously flips to the correct answer. A n-qubit quantum computer is not like 2^n phantom computers running at the same time in some quantum superposition phantom-zone. That’s the popular misconception, but it’s effectively ignorant techno-woo.

Here’s what really happens. If you have a string of n-qubits, when you measure them, they might end up randomly in [one] of the 2^n possible configurations. However, if you apply some operations to your string of n-qubits using quantum gates, you can usefully bias their wave equations, such that the probabilities of certain configurations are much more likely to appear. (You can’t have too many of these operations, however, as that runs the risk of decoherence.) Hopefully, you can do this in such a way, that the biased configurations are the answer to a problem you want to solve.

So then, if you have a quantum computer in such a setup, you can run it a bunch of times, and if everything goes well after enough iterations, you will be able to notice a bias towards certain configurations of the string of bits. If you can do this often enough to get statistical significance, then you can be pretty confident you’ve found your answers.

— An Argument Against Quantum Computers

— Hacker News

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2018.05.17 Thursday ACHK

Van der Waals equation 1.2

Whether X_{\text{measured}} is bigger or smaller than X_{\text{ideal}} ultimately depends on the assumptions and definitions used in the derivation of the ideal gas equation itself.

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In the ideal gas equation derivation, the volume used in the equation refers to the volume that the gas molecules can move within. So

V_{\text{ideal}} = V_{\text{available for a real gas' molecules to move within}}

Then, when deriving the pressure, it is assumed that there are no intermolecular forces among gas molecules. So

P_{\text{ideal}} = P_{\text{assuming no intermolecular forces}}

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These are the reasons that

V_{\text{ideal}} < V_{\text{measured}}

P_{\text{ideal}} > P_{\text{measured}}

P_{\text{ideal}} V_{\text{ideal}} = nRT

\left(P_\text{measured} + a\left(\frac{n}{V}\right)^2\right) \left(V_\text{measured}-nb\right) = nRT

— Me@2018-05-16 07:12:51 PM

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… the thing to keep in mind is that the “pressure we use in the ideal gas law” is not the pressure of the gas itself. The pressure of the gas itself is too low: to relate that pressure to “pressure for the ideal gas law” we have to add a number. While the volume occupied by the real gas is too large – the “ideal volume” is less than that. – Floris Sep 30 ’16 at 17:34

— Physics Stackexchange

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2018.05.16 Wednesday (c) All rights reserved by ACHK

Tree rings, 2

69b73-growth_rings

This file is licensed under the Creative Commons Attribution 2.0 Generic license. Author: Lawrence Murray from Perth, Australia

Time-traveling to the past is like “making an outside ring more inside”, which is logically impossible.

— Me@2011.09.18

8fd2c-pastpresentfuture

Me@2010

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2018.05.16 Wednesday (c) All rights reserved by ACHK

Richard Stallman

I’ve always lived cheaply. I live like a student, basically. And I like that, because it means that money is not telling me what to do. I can do what I think is important for me to do. It freed me to do what seemed worth doing. So make a real effort to avoid getting sucked into all the expensive lifestyle habits of typical Americans. Because if you do that, then people with the money will dictate what you do with your life. You won’t be able to do what’s really important to you.

— Richard Stallman

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There is nothing wrong to be a student-having a lot of new learnings and new young friends, as long as you can earn enough money.

— Me@2011.08.20

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2018.05.15 Tuesday (c) All rights reserved by ACHK

On Keeping Your Soul, 3

這段改編自 2010 年 4 月 18 日的對話。

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「學術研究的論文」和「網誌文章」的主要分別是,論文要花很多時間,才可寫成一篇,然後只有極少人去閱讀:

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時間 ~ 1 至 2 年

潛在讀者人數 ~ 10 至 100 人

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網誌文章則相反:

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時間 ~ 1 至 2 星期

潛在讀者人數 ~ > 100 人

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不過,曲高自然和寡:

如果一篇網誌文章的學術成份高的話,閱讀人數自然極少。

— Me@2018-05-15 05:00:47 PM

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後記:

英雄所見略同,物理學家 Lubos Motl,有近乎相同的見解。

— Me@2018-05-15 05:00:55 PM

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2018.05.15 Tuesday (c) All rights reserved by ACHK