Reading Russian Lines of Cloud Atlas (4)

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ОБЛАЧНЫЙ АТЛАС

Острова Тихого Океана, 1849 год.

Дело сделано, мистер Юинг, теперь сей контракт свят для исполнения… почти как Десять Заповедей.

сделано: done
сей: this
свят: holy
исполнения: execution, fulfillment
почти: almost
Заповедей: commandment

Благодарю, Преподобный Хоррекс, мой тесть с нетерпением ждёт завершения этой сделки.
Преподобный: Reverend
тесть: father-in-law
нетерпением: impatience, hurry
ждёт: wait for, expect
завершения: completion
сделки: transaction Continue reading “Reading Russian Lines of Cloud Atlas (4)”

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Reading Russian Lines of Cloud Atlas (3)

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– Здравствуйте.
– Ваш пропуск.
Смотрю вы начеку?

пропуск: pass, admission
начеку: on the alert


Я был издателем Дермонта Хоггинса, а не психоаналитиком или астрологом, и это чёртова, чёртова правда, я и понятия не имел, что он сделает в тот вечер.

издателем: publisher
психоаналитиком: psychoanalyst
астрологом: astrologer
понятия: idea
имел: have
тот: that
вечер: evening

Continue reading “Reading Russian Lines of Cloud Atlas (3)”

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Quotes From Cloud Atlas (I)

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Теперь я понимаю, что границы между шумом и звуком условный. Любые границы условный, и созданный, чтобы их переступать. Всё условности преодолимый, стоит лишь поставить для себя эту цель.

граница: limit, confine
между: between
шум: noise Continue reading “Quotes From Cloud Atlas (I)”

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Polynomials and Power Series (I)

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Today we discuss something on polynomials.

Over a Commutative Ring

Nilpotents and units are closely related. In a commutative unital ring R, if x nilpotent, a unit, then a+x is again a unit. If 1+x y is a unit for every y\in R, then x\in\mathfrak{R}, the Jacobson radical, approximately nilpotent.
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Reading Russian Lines of ‘Cloud Atlas’ (2)

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Вот тогда-то я и свёл знакомство с доктором Генри Гуссом, с человеком, который, как я надеялся, сможет излечить мой недуг.
Вы что-то ищите?


тогда-то: then, at that time
свёл (свести): took; свёл знакомство с: made the acquaintance of
который: who
надеялся: hope
сможет: be able to
излечить: cure
недуг: ailment, illness
что-то: something
ищите: look for, search after

Continue reading “Reading Russian Lines of ‘Cloud Atlas’ (2)”

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Discussion on Exercises of Commutative Algebra (I)

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  1. Units
    , nilpotents, and idempotents lift from A/\mathfrak{N} to A.Proof: Units and nilpotents are obvious. In fact they lift to any of their representatives.
    For idempotents, if x^2=x\in A/\mathfrak{N}, then (1-x)x=0 \in A/\mathfrak{N}, so (1-x)^kx^k=0\in A for sufficiently large k. And (1-x)^k+x^k=1-x+x=1\in A/\mathfrak{N}, so lifts to a unit (1-x)^k+x^k. Moreover, its inverse u=1\in A/\mathfrak{N}. So (ux)^k(u(1-x))^k=0,ux^k+u(1-x)^k=1\in A and ux=x,u(1-x)=1-x\in A/\mathfrak{N}.
    This can be interpreted by sheaf theory, which is to be discussed in later posts. Continue reading “Discussion on Exercises of Commutative Algebra (I)”
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A Simple Combinatorial Problem and Related Thoughts

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Problem: If the decimal expansion of a contains all 10 digits (0,...,9), then the number of length n strings (shorted as n-strings) is greater than n+8.

If you’ve established the simple lemma, the solution is instant. Otherwise very impossible.

Lemma: The number C_n of n-strings is strictly greater than C_{n-1}, that of n-1-strings.
Actually,  we always have C_n \ge C_{n-1}: Every n-1-string corresponds to an n-string by continuing 1 more digit. The map is clearly injective. If C_n=C_{n-1}, it is bijective, meaning we have a way to continue uniquely, which means rationality. Rigidly, at least one of the n-1-strings occurs infinitely, but all digits after some n-1-string is totally determined by it. So if an n-1-string appears twice, it must appear every such distances, and so do the digits between. Continue reading “A Simple Combinatorial Problem and Related Thoughts”

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Generating Functions and Formal Power Series

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Today we discuss some generating functions.

Problem 1: Give a finite set of positive integers T. Let \mathfrak{T}_n be the collection of sequences (t_1,t_2,...,t_m), such that \sum_{i=1}^m t_m=n, and each t_i\in T. Let a_n=|\mathfrak{T}_n| for n\ge1 and a_0=1, and f(x)=\sum_{n=0}^{\infty}a_n x^n. Find out what is f(x) explicitly.

Solution: It is not hard to note the recursive relation a_n=\sum_{t\in T} a_{n-t} for n\ge1, if we set a_i=0 for negative i. So that f(x)=1+\sum_{t\in T} x^t f(x) and f(x)=1/(1-\sum_{t\in T} x^t), which is a rational function.

Variantion 1: If T is infinite, what would happen? Would f(x) still be rational?

We first analyze simple cases. If T=\mathbb{N}^+, it is expected that f(x)=1+\sum_{t=1}^{\infty} x^t f(x)=1+f(x) x/(1-x), so that f(x)=(1-x)/(1-2x)=1+\sum_{n=1}^{\infty} 2^{n-1} x^n. Indeed, in this case, counting the sequences amounts to divide a sequence of n objects arbitrarily. You can choose to divide between the ith and i+1th for 1\ge i\ge n-1, so in all 2^{n-1} choices, justifying the expansion.

I think it is equivalent to f being rational.

Theorem: \mathbb{Q}_p(t)\cap\mathbb{Q}[[t]]=\mathbb{Q}(t)

It is very interesting that this theorem is used for the rationality of \zeta-functions for algebraic varieties, which is part of the Weil conjectures.

Wednesday, July 10, 2013

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Reading Russian Lines of ‘Cloud Atlas’ (1)

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Тоскливая ночь.
Обыватели печальный, ветер пробирает до костей.
В нём я слышу…голоса.
Это вой, вой предков, они рассказывают свои истории.
Их голоса сплетаются в хор.
Но один голос особенный…
Этот голос, шепчет, преследует тебя из мрака.
Клыкастый дьявол, сам Старина Джорджи.
Приготовитесь слушать, и я расскажу тебе о том, как мы встретились в первый раз, лицом к лицу.




Words:

Тоскливая: depressed, sad
ночь: night

Continue reading “Reading Russian Lines of ‘Cloud Atlas’ (1)”

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A Quote From ‘Cloud Atlas’

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На улице тяжёлый снег падал на шиферные крыши и гранитные стены. Подобно Солженицыну, томившемуся в Вермонте,
я буду трудиться
в изгнании.
Но в отличие от Солженицына, я буду не один.



Explanation:

green: radical
red: prefix
blue: suffix
dark red: preposition
orange: auxiliary ingredients
purple: declension
grey
background
: accent

 

 

underline: proper noun
Friday, July 12, 2013
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Faith

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When I was a child, I was attracted by the world beyond our planet. I did not have the opportunity to see a really starry night, for the I did not often live in the country side. But I could still imagine. I bought I lot of books which introduced efforts humans had made on exploring the outer world. I also watch cartoons related to this topic. My yearning for the vast world first led me to science.

Years have passed. Now I’ve been concentrating on mathematics. But the deep spaces still play an important role in my life. I am not fond of money, honor, reputation or something else, for I believe that there is something far greater. Money, honor or reputation only works in human society and the duplicity, deception and disguise intended for these things are simply nothing but abjection in front of the universe. Continue reading “Faith”

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An Animation and the Middle Age

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1

This weekend I came across the anime Spice and Wolf. The character Horo is so cute, but this is not what I want to mention most.

It was in the Middle Age, Lawrence, the main character is a travelling merchant. Therefore trading activities and religious organizations(sorry I can’t find an equivalent word) are often mentioned in the story. That inspired me to the imagination of the Middle Age.

It was a dark age that religious power controlled the society, and however, an age of merchants. There were villages, towns, cities, ports and kingdoms. There were missionaries, knights, mercenaries and kings. It was hard to transmit information. It was hard to transport substances. It was a solid age. And it was merchants, who traveled everywhere for more benefits, that liquefied the age. Continue reading “An Animation and the Middle Age”

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1

Diary (Sep 4, 2010)

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The book From Calculus to Cohomology by Madsen looks a nice book. Well, it’s the first time I saw an introduction to topology from CALCULUS so I believe it amazing. Maybe I am too ignorant in mathematics.

I’m studying Commutative Algebra and Topology recently. As far as I’m concerned, categories are fundamental in mathematics so I’m particularly fond of algebra in favor of the concept of category. I saw some one says geometry and topology structures are much more interesting than algebra structures. I don’t understand it very much. Maybe in my mind all structures are algebraic. Well, maybe when we adopt measures and Cauchy sequences we are entering the field of Analysis, Geometry and Topology? Actually in my heart it is still algebraic. I think continuity doesn’t mean non-algebra. Continue reading “Diary (Sep 4, 2010)”

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How do we think?

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When I say “apple” ,an apple will appear in your mind. You can understand it. People may explain it as some correspondence between images and specific signs. And the signs themselves are images so actually we associate the images with sounds and then sounds with signs. Humans have evolved so much that today they can understand written language in silence, not needing to read them out.

The understand of concrete things may be as easy as the above. But what about that of abstract things, concepts and statements? That should be hard to explain because explanation itself is an abstract statement, which I must use my understanding system of abstract things to understand.

I used to guess, the approach to solve this problem is up to science. That is, to study the brain. But there is still a critical obstacle that we still have to think using our brain.

Is that the unreachable acme of human cognition? Do humans have to continue evolving to answer it? Or never?

2010-09-04

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A Testing Post

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Human and Natural Disasters

Human activities are largely involved in the formation of some natural disasters, while not involved in the formation of some others.

As we all know, it is obvious that humans have nothing to do with the formation of volcano eruptions or earthquakes. But some other disasters, for instance, sandstorms, are related to human activities. Overgrazing and overcutting ruin vegetation, in turn causing desertification. When sands meet a windstorm, a sandstorm comes into being. Another instance is acid rain. Waste gas produced in industry activities, mostly in thermal powering, goes up to the clouds. It then blends with the water vapors. As a result, the rain from the clouds becomes acid, doing a lot of harm to humans, animals, plants, etc. Continue reading “A Testing Post”

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