Renowned Chinese mathematician Wu Wenjun dies at 98

Source: https://news.cgtn.com/news/3d517a4e33637a4d/share_p.html

Wu Wenjun, distinguished mathematician, member of the Chinese Academy of Sciences (CAS), and winner of China’s Supreme Scientific and Technological Award winner, died at the age of 98 on Sunday in Beijing, according to the CAS.
Wu was born in Shanghai on May 12, 1919. In 1940, he graduated from Shanghai Jiao Tong University, and received a PhD from the University of Strasbourg, France in 1947.
In 1951, Wu returned to China and served as a math professor at Peking University. He made great contributions to the field of topology by introducing various principles now recognized internationally.
In the field of mathematics mechanization, Wu suggested a computerized method to prove geometrical theorems, known as Wu’s Method in the international community.
He was elected as a member of the CAS in 1957 and as a member of the Third World Academy of Sciences in 1990.
Wu Wenjun was given China’s Supreme Science and Technology Award by the then President Jiang Zemin in 2000, when this highest scientific and technological prize in China began to be awarded.

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Rise in WordAds earnings March 2017

WordPress and WordAds have been doing a good job, I must say. In this year (2017), the earnings have increased gradually for my site (view count is approximately constant).

Certainly a great improvement from 2016, where I got a measly $0.70 for 11824 ad impressions in one month.

Keep it up, WordPress!

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Ye Sons and Daughters (O Filii et Filiae)

Nice traditional hymn.

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What is a Module (模)? 

Math Online Tom Circle

Replace Field scalar as in a Vector Space to Ring scalar in a Module.

Module is more powerful than Vector Space – because Ring has an “Ideal” (理想) which can partition it to Quotient Ring, but Field (scalar in a Vector Space) can’t.

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American Genius Series 1 1of8 Jobs vs Gates (2015)

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Structure Preserving

Math Online Tom Circle

Abstract Algebra studies all kinds ofmathematical structures(Group, Ring, Vector Space, Category, …) , and relationship between them if the structures are preserved after mapping.

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The Cry of the Poor by John Michael Talbot

Recently I discovered a classical guitar Christian composer on YouTube – John Michael Talbot. His music is quite amazing, and one of a kind.

  1. The Cry of the Poor

2. Be not afraid

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Five Loaves and Two Fishes – Corrinne May (Illustrated)

Corrinne May is a Singaporean singer.

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Mathematics: Beauty vs Utility – Numberphile

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2 minutes pour le théorème de Bézout

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Maths -Topologie

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How the Staircase Diagram changes when we pass to derived couple (Spectral Sequence)

Set A_{n,p}^1=H_n(X_p) and E_{n,p}^1=H_n(X_p,X_{p-1}). The diagram then has the following form:

When we pass to the derived couple, each group A_{n,p}^1 is replaced by a subgroup A_{n,p}^2=\text{Im}\,(i_1: A_{n,p-1}^1\to A_{n,p}^1). The differentials d_1=j_1k_1 go two units to the right, and we replace the term E_{n,p}^1 by the term E_{n,p}^2=\text{Ker}\, d_1/\text{Im}\,d_1, where the d_1‘s refer to the d_1‘s leaving and entering E_{n,p}^1 respectively.

The maps j_2 now go diagonally upward because of the formula j_2(i_1a)=[j_1a]. The maps i_2 and k_2 still go vertically and horizontally, i_2 being a restriction of i_1 and k_2 being induced by k_1.

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How do you overcome the fear of the future? Pope Francis provides the keys

Source: http://aleteia.org/2017/04/26/how-do-you-overcome-the-fear-of-the-future-pope-francis-provides-the-keys/

Have you ever been gripped by a fear of the future that left you walking in circles or paralyzed from moving forward? Today, Pope Francis provided a deeper answer to this common human fear, urging Christians to remember that faith is the anchor that keeps our lives moored to the heart of God, amid every storm and difficulty.

Speaking to faithful and pilgrims at today’s general audience in St. Peter’s Square, the pope reminded them that, wherever we go, God’s love goes before us.

“There will never be a day in our lives when we will cease being a concern for the heart of God,” he said, as he continued his series on Christian hope.

“I am with you”

Reflecting on St. Matthew’s Gospel, Pope Francis observed that it begins with the birth of Jesus as Emmanuel — “God is with us” — and concludes with the Risen Lord’s promise to his disciples: “I am with you always, even to the end of the age” (Mt 28:20). He said:

The whole gospel is enclosed in these two quotations, words that communicate the mystery of a God whose name, whose identity is ‘to be with,’ in particular, to be with us, with the human creature. Ours is not an absent God … He is a God who is “passionately” in love with man, and so tender a lover that he is incapable of separating Himself from him. We humans are able to break bonds and bridges. He is not. If our heart cools, his remains incandescent. Our God always accompanies us, even if we unfortunately forgot about Him. […] Christians especially do not feel abandoned, because Jesus promises not to only wait for us at the end of our long journey, but to accompany us during each of our days.

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Relative Homology Groups

Given a space X and a subspace A\subset X, define C_n(X,A):=C_n(X)/C_n(A). Since the boundary map \partial: C_n(X)\to C_{n-1}(X) takes C_n(A) to C_{n-1}(A), it induces a quotient boundary map \partial: C_n(X,A)\to C_{n-1}(X,A).

We have a chain complex \displaystyle \dots\to C_{n+1}(X,A)\xrightarrow{\partial_{n+1}}C_n(X,A)\xrightarrow{\partial_n}C_{n-1}(X,A)\to\dots where \partial^2=0 holds. The relative homology groups H_n(X,A) are the homology groups \text{Ker}\,\partial_n/\text{Im}\,\partial_{n+1} of this chain complex.

Relative cycles
Elements of H_n(X,A) are represented by relative cycles: n– chains \alpha\in C_n(X) such that \partial\alpha\in C_{n-1}(A).

Relative boundary
A relative cycle \alpha is trivial in H_n(X,A) iff it is a relative boundary: \alpha=\partial\beta+\gamma for some \beta\in C_{n+1}(X) and \gamma\in C_n(A).

Long Exact Sequence (Relative Homology)
There is a long exact sequence of homology groups:
\begin{aligned}  \dots\to H_n(A)\xrightarrow{i_*}H_n(X)\xrightarrow{j_*}H_n(X,A)\xrightarrow{\partial}H_{n-1}(A)&\xrightarrow{i_*}H_{n-1}(X)\to\dots\\  &\dots\to H_0(X,A)\to 0.  \end{aligned}

The boundary map \partial:H_n(X,A)\to H_{n-1}(A) is as follows: If a class [\alpha]\in H_n(X,A) is represented by a relative cycle \alpha, then \partial[\alpha] is the class of the cycle \partial\alpha in H_{n-1}(A).

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Prayer of St Thomas Aquinas for Students (English and Latin)

Source: http://www.aquinascollege.edu/prayer-before-study-exams-spring-2013/

A Prayer Before Study

Ineffable Creator,
Who, from the treasures of Your wisdom,
have established three hierarchies of angels,
have arrayed them in marvelous order
above the fiery heavens,
and have marshaled the regions
of the universe with such artful skill,

You are proclaimed
the true font of light and wisdom,
and the primal origin
raised high beyond all things.

Pour forth a ray of Your brightness
into the darkened places of my mind;
disperse from my soul
the twofold darkness
into which I was born:
sin and ignorance.

You make eloquent the tongues of infants.
refine my speech
and pour forth upon my lips
The goodness of Your blessing.

Grant to me
keenness of mind,
capacity to remember,
skill in learning,
subtlety to interpret,
and eloquence in speech.

May You
guide the beginning of my work,
direct its progress,
and bring it to completion.

You Who are true God and true Man,
who live and reign, world without end.

Amen.

Ante Studium

Creator ineffabilis,
qui de thesauris sapientiae tuae
tres Angelorum hierarchias designasti,
et eas super caelum empyreum
miro ordine collocasti,
atque universi partes elegantissime disposuisti,

tu inquam qui
verus fons
luminis et sapientiae diceris
ac supereminens principium

infundere digneris
super intellectus mei tenebras
tuae radium claritatis,
duplices in quibus natus sum
a me removens tenebras,
peccatum scilicet et ignorantiam.

Tu, qui linguas infantium facis disertas,
linguam meam erudias
atque in labiis meis gratiam
tuae benedictionis infundas.

Da mihi
intelligendi acumen,
retinendi capacitatem,
addiscendi modum et facilitatem,
interpretandi subtilitatem,
loquendi gratiam copiosam.

Ingressum instruas,
progressum dirigas,
egressum compleas.

Tu, qui es verus Deus et homo,
qui vivis et regnas in saecula saeculorum.

Amen.

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Exact sequence (Quotient space)

Exact sequence (Quotient space)
If X is a space and A is a nonempty closed subspace that is a deformation retract of some neighborhood in X, then there is an exact sequence
\begin{aligned}  \dots\to\widetilde{H}_n(A)\xrightarrow{i_*}\widetilde{H}_n(X)\xrightarrow{j_*}\widetilde{H}_n(X/A)\xrightarrow{\partial}\widetilde{H}_{n-1}(A)&\xrightarrow{i_*}\widetilde{H}_{n-1}(X)\to\dots\\  &\dots\to\widetilde{H}_0(X/A)\to 0  \end{aligned}
where i is the inclusion A\to X and j is the quotient map X\to X/A.

Reduced homology of spheres (Proof)
\widetilde{H}_n(S^n)\cong\mathbb{Z} and \widetilde{H}_i(S^n)=0 for i\neq n.

For n>0 take (X,A)=(D^n,S^{n-1}) so that X/A=S^n. The terms \widetilde{H}_i(D^n) in the long exact sequence are zero since D^n is contractible.

Exactness of the sequence then implies that the maps \widetilde{H}_i(S^n)\xrightarrow{\partial}\widetilde{H}_{i-1}(S^{n-1}) are isomorphisms for i>0 and that \widetilde{H}_0(S^n)=0. Starting with \widetilde{H}_0(S^0)=\mathbb{Z}, \widetilde{H}_i(S^0)=0 for i\neq 0, the result follows by induction on n.

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Reduced Homology

Define the reduced homology groups \widetilde{H}_n(X) to be the homology groups of the augmented chain complex \displaystyle \dots\to C_2(X)\xrightarrow{\partial_2}C_1(X)\xrightarrow{\partial_1}C_0(X)\xrightarrow{\epsilon}\mathbb{Z}\to 0 where \epsilon(\sum_i n_i\sigma_i)=\sum_in_i. We require X to be nonempty, to avoid having a nontrivial homology group in dimension -1.

Relation between H_n and \widetilde{H}_n
Since \epsilon\partial_1=0, \epsilon vanishes on \text{Im}\,\partial_1 and hence induces a map \tilde{\epsilon}:H_0(X)\to\mathbb{Z} with \ker\tilde{\epsilon}=\ker\epsilon/\text{Im}\,\partial_1=\widetilde{H}_0(X). So H_0(X)\cong\widetilde{H}_0(X)\oplus\mathbb{Z}. Clearly, H_n(X)\cong\widetilde{H}_n(X) for n>0.

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8 JCs to merge (i.e. 4 JCs to close down)

The latest education news in Singapore is that 4 pairs of JCs to merge as student numbers shrink; 14 primary and 6 secondary schools also affected.

The effect on the primary and secondary schools is not that significant, due to the large number of primary and secondary schools. However, there are only around 20 JCs in Singapore, the effect is quite big for JCs.

8 JCs merging is just a nice way of saying 4 JCs to be shut down permanently. RIP Serangoon, Tampines, Innova and Jurong JCs.

The most affected would be O level students in the next 5 years. Yes, there is declining birthrate but that is gradual. So for the next 5 years, there is approximately the same number of students competing for 4 less JCs.

So by “Demand and Supply” logic, we have:
– similar demand for JCs (approx. same number of students in the next 5 years)
– lower supply of JCs (due to the 4 axed JCs)

By Economic Theory: If supply decreases and demand is unchanged, then it leads to a higher equilibrium price.

Hence the logical conclusion is that the “price” will rise, that is, cutoff points for JCs may become lower. To add on to that, the 4 axed JCs cater mainly to the 13-20 pointers. So students falling in that L1R5 range will be especially affected.


Also check out: Which JC is good?

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中国初中生集体挑战美国高考数学SAT题

Math Online Tom Circle

[Hint]:

$latex (a + x)^{frac {1}{3}} + (a – x)^{frac {1}{3}} = 2(a)^{frac {1}{3}} + 2(x)^{frac {1}{3}} $

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The World’s Best Mathematician (*)

Math Online Tom Circle

Terence Tao:

  • Weakness in Math: Algebraic Topology
  • Collaborative & “Lone Wolf”
  • Not attempting the Riemann Hypothese: tools not there yet.


3 Phases of Math Training:

  1. Pre-rigourous: intuition
  2. Rigorous: formal prove
  3. Post-rigourous: 1 + 2

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The Legend of Question Six (IMO 1988)

Math Online Tom Circle

This question was submitted by West Germany to the IMO Committee, the examiners could not solve it in 6 hours.

In the IMO (1988) only 11 contestants solved it, one of them proved it elegantly. Terence Tao (Australia) only got 1 mark out of 7 in this question.

Solution:

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The Shortest Ever Papers 

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Klein Bottle as Gluing of Two Mobius Bands

This is a nice picture on how the Klein bottle can be formed by gluing two Mobius bands together. Very neat and self-explanatory!

Source: https://math.stackexchange.com/questions/907176/klein-bottle-as-two-m%C3%B6bius-strips

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Mayer-Vietoris Sequence applied to Spheres

Mayer-Vietoris Sequence
For a pair of subspaces A,B\subset X such that X=\text{int}(A)\cup\text{int}(B), the exact MV sequence has the form
\begin{aligned}  \dots&\to H_n(A\cap B)\xrightarrow{\Phi}H_n(A)\oplus H_n(B)\xrightarrow{\Psi}H_n(X)\xrightarrow{\partial}H_{n-1}(A\cap B)\\  &\to\dots\to H_0(X)\to 0.  \end{aligned}

Example: S^n
Let X=S^n with A and B the northern and southern hemispheres, so that A\cap B=S^{n-1}. Then in the reduced Mayer-Vietoris sequence the terms \tilde{H}_i(A)\oplus\tilde{H}_i(B) are zero. So from the reduced Mayer-Vietoris sequence \displaystyle \dots\to\tilde{H}_i(A)\oplus\tilde{H}_i(B)\to\tilde{H}_i(X)\to\tilde{H}_{i-1}(A\cap B)\to\tilde{H}_{i-1}(A)\oplus\tilde{H}_{i-1}(B)\to\dots we get the exact sequence \displaystyle 0\to\tilde{H}_i(S^n)\to\tilde{H}_{i-1}(S^{n-1})\to 0.
We obtain isomorphisms \tilde{H}_i(S^n)\cong\tilde{H}_{i-1}(S^{n-1}).

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Resucitó (Spanish Easter Song)

Wishing all readers a happy Easter!

This is a famous Spanish Easter Song: Resucitó, with translation.

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Spectral Sequence

Spectral Sequence is one of the advanced tools in Algebraic Topology. The following definition is from Hatcher’s 5th chapter on Spectral Sequences. The staircase diagram looks particularly impressive and intimidating at the same time.

Unfortunately, my LaTeX to WordPress Converter app can’t handle commutative diagrams well, so I will upload a printscreen instead.

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Holy, Holy, Holy Lord: Lyrics and Music

Amazing organ music.

Lyrics:

Holy, holy, holy Lord,

God of power and (God of) might

Heaven and earth are full of Your glory

Hosanna in the highest

Blessed is He who comes in the name of the Lord

Hosanna in the highest, Hosanna in the highest.

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Real World Applications of Algebra, Geometry and Topology

Quite a nice video here:

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Love Jesus in all who suffer, pope says on Palm Sunday

Source: https://cnstopstories.com/2017/04/09/love-jesus-in-all-who-suffer-pope-says-on-palm-sunday/

By Carol Glatz
Catholic News Service

VATICAN CITY (CNS) — Jesus does not ask that people only contemplate his image, but that they also recognize and love him concretely in all people who suffer like he did, Pope Francis said.

Jesus is “present in our many brothers and sisters who today endure sufferings like his own — they suffer from slave labor, from family tragedies, from diseases. They suffer from wars and terrorism, from interests that are armed and ready to strike,” the pope said April 9 as he celebrated the Palm Sunday Mass of the Lord’s Passion.

In his noon Angelus address, the pope also decried recent terrorist attacks in Sweden and Egypt, calling on “those who sow terror, violence and death,” including arms’ manufacturers and dealers, to change their ways.

In his prayers for those affected by the attacks, the pope also expressed his deepest condolences to “my dear brother, His Holiness Pope Tawadros, the Coptic church and the entire beloved Egyptian nation,” which the pope was scheduled to visit April 28-29.

At least 15 people were killed and dozens more injured April 9 in an Orthodox church north of Cairo as Coptic Christians gathered for Palm Sunday Mass; the attack in Sweden occurred two days earlier when a truck ran through a crowd outside a busy department store in central Stockholm, killing four and injuring 15 others.

The pope also prayed for all people affected by war, which he called, a “disgrace of humanity.”

Tens of thousands of people carrying palms and olive branches joined the pope during a solemn procession in St. Peter’s Square under a bright, warm sun for the beginning of Holy Week.

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OneKey Token Out of Battery — What to do?

It seems like the OneKey Token (used for 2 factor authentication) runs out of battery quite fast. I have barely used mine (usually use SMS), yet its battery has died surprisingly soon, compared to my other bank tokens.

What should one do in this case? Please comment below if you have other options.

Once the battery goes to zero, it appears that there are only two options:

Option 1)

Go to their office PSA Building (Alexandra Road) or International Plaza (Anson Road), to get free** replacement (**provided still within the warranty period of 1 year).

Waiting time is estimated 40-50 minutes. (and “free” may not be guaranteed, depends on whether you meet their requirement of warranty period, etc.)

Disclaimer: I have not tried this method myself. This is based on the Hardwarezone post linked below.

Option 2)

Go to this OneKey Assurity site https://portal.assurity.sg/naf-web/public/index.do to purchase a new token at $15. (Quite expensive 😦 )

Source: http://forums.hardwarezone.com.sg/money-mind-210/one-key-token-low-battery-5593102.html

I have no idea why the battery runs out so fast. Even my cheap Casio watch’s battery (which is running 24 hours a day) lasts longer than this token’s (which I have pressed less than 10 times).

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Unknown German Retiree Proved The “Gaussian Correlation Inequality” Conjecture

Math Online Tom Circle

https://www.wired.com/2017/04/elusive-math-proof-found-almost-lost

$latex boxed {P (a + b) geq P (a) times P (b)}&fg=aa0000&s=3$

Case “=” : if (a, b) independent
Case “>” : if (a, b) dependent

Thomas Royen used only high-school math (function, derivative) in his proof in 2014. He then published it in arxiv.org website – likePerelman did with the “Poincaré Conjecture”.

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Tough Math : Diophantine Equation

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Fun Math

Math Online Tom Circle

Answer : (scroll below)

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Answer = 3 mins

See clearer if change person to taxi car, bun to passenger.

9 taxi cars send 9 passengers will take the SAME timing as 3 taxi cars send 3 passengers.

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Category Adjunctions 伴随函子

Math Online Tom Circle

Adjunction is the “weakeningof Equality” in Category Theory.

Equivalence of 2 Categories if:

5.2 Adjunction definition: $latex (L, R, epsilon, eta )$ such that the 2 triangle identities below ( red and blue) exist.

6.1Prove: Let C any category, D a Set.

$latex boxed {text {C(L 1, -)} simeq text {R}}&fg=aa0000&s=3$

$latex {text {Right Adjoint R in Set category is }}&fg=aa0000&s=3$ $latex {text {ALWAYS Representable}}&fg=aa0000&s=3$

1 = Singleton in Set D

From an element in the singleton Set always ‘picks’ (via the function) an image element from the other Set, hence :
$latex boxed {text{Set (1, R c) } simeq text{Rc }}&fg=0000aa&s=3$

Examples : Product & Exponential are Right Adjoints

Note: Adjoint is a more powerful concept to understand than the universal construction of Product and Exponential.

Adjoint (伴随) Functors (函子):

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Category Theory (Stanford Encyclopedia of Philosophy)

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Beautiful Math Olympiad Problem

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The genius who rejected Fields Medal & Clay Prize: Grigori Perelman 

Math Online Tom Circle

Russian mathematician Grigori Perelman proved the Poincaré Conjecture in 7 years of solitude research in his Russian apartment - same 7 years of solitude forAndrew Wiles (The Fermat’s Last Theorem)in the Cambridge attic house andZhang Yitang 张益唐 (7-Million-Gap Twin Primes) in the “Subway” sandwich kitchen.

7 is the Perfect Number. 1 week has 7 days, according to the “Book of Genesis”, God created the universe in 6 days and rested “Sabbath” on the beautiful 7th day.

People involved in his journey:

  • His mother who supported his early education in Math, all the way to his international fame and adulthood —Erdös Paulalso had a mother supporting her single son in the entire Mathematical life.
  • Prof Hamilton who “unintentionally”gave him the inspiration of the “Ricci-flow” tool – the first key to the door of The Poincaré Conjecture.
  • His closed Chinese friend the MIT…
  • View original post 76 more words

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    The Yoneda Lemma

    Math Online Tom Circle

    Representable Functor F of C ( a, -):

    $latex boxed {(-)^{a} = text {F} iff a = text {log F}}&fg=aa0000&s=3$

    4.2Yoneda Lemma

    Prove :

    Yoneda Lemma:
    $latex text {F :: C} to text {Set}$

    $latex boxed {alpha text { :: [C, Set] (C (a, -),F) } simeq text {F a}}&fg=0000aa&s=3$

    $latex alpha : text {Natural Transformation}$
    $latex simeq : text {(Natural) Isomorphism}$

    Proof: By “Diagram chasing” below, shows that

    Left-side
    : $latex alpha text { :: [C, Set] (C (a, -),F) } $ is indeed a (co-variant) Functor.

    Right-side: Functor “F a“.

    Note: When talking about the natural transformations, always mention their component “x”: $latex alpha_{x}, beta_{x}$

    Yoneda Embedding (Lagatta)

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    SO(3) diffeomorphic to RP^3

    SO(3)\cong\mathbb{R}P^3}

    Proof:

    We consider SO(3) as the group of all rotations about the origin of \mathbb{R}^3 under the operation of composition. Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation.

    We consider \mathbb{R}P^3 as the unit 3-sphere S^3 with antipodal points identified.

    Consider the map from the unit ball D in \mathbb{R}^3 to SO(3), which maps (x,y,z) to the rotation about (x,y,z) through the angle \pi\sqrt{x^2+y^2+z^2} (and maps (0,0,0) to the identity rotation). This mapping is clearly smooth and surjective. Its restriction to the interior of D is injective since on the interior \pi\sqrt{x^2+y^2+z^2}<\pi. On the boundary of D, two rotations through \pi and through -\pi are the same. Hence the mapping induces a smooth bijective map from D/\sim, with antipodal points on the boundary identified, to SO(3). The inverse of this map, \displaystyle ((x,y,z),\pi\sqrt{x^2+y^2+z^2})\mapsto(x,y,z) is also smooth. (To see that the inverse is smooth, write \theta=\pi\sqrt{x^2+y^2+z^2}. Then x=\sqrt{\frac{\theta^2}{\pi^2}-y^2-z^2}, and so \frac{\partial^k x}{\partial\theta^k} exists and is continuous for all orders k. Similar results hold for the variables y and z, and also mixed partials. By multivariable chain rule, one can see that all component functions are indeed smooth, so the inverse is smooth as claimed.)

    Hence SO(3)\cong D/\sim, the unit ball D in \mathbb{R}^3 with antipodal points on the boundary identified.

    Next, the mapping \displaystyle (x,y,z)\mapsto (x,y,z,\sqrt{1-x^2-y^2-z^2}) is a diffeomorphism between D/\sim and the upper unit hemisphere of S^3 with antipodal points on the equator identified. The latter space is clearly diffeomorphic to \mathbb{R}P^3. Hence, we have shown \displaystyle SO(3)\cong D/\sim\cong\mathbb{R}P^3.

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    SU(2) diffeomorphic to S^3 (3-sphere)

    SU(2)\cong S^3 (diffeomorphic)

    Proof:
    We have that \displaystyle SU(2)=\left\{\begin{pmatrix}\alpha &-\overline{\beta}\\  \beta &\overline{\alpha}\end{pmatrix}: \alpha, \beta\in\mathbb{C}, |\alpha|^2+|\beta|^2=1\right\}.

    Since \mathbb{R}^4\cong\mathbb{C}^2, we may view S^3 as \displaystyle S^3=\{(\alpha,\beta)\in\mathbb{C}^2: |\alpha|^2+|\beta|^2=1\}.

    Consider the map
    \begin{aligned}f: S^3&\to SU(2)\\  f(\alpha,\beta)&=\begin{pmatrix}\alpha &-\overline{\beta}\\  \beta &\overline{\alpha}\end{pmatrix}.  \end{aligned}

    It is clear that f is well-defined since if (\alpha,\beta)\in S^3, then f(\alpha,\beta)\in SU(2).

    If f(\alpha_1,\beta_1)=f(\alpha_2,\beta_2), it is clear that (\alpha_1, \beta_1)=(\alpha_2,\beta_2). So f is injective. It is also clear that f is surjective.

    Note that SU(2)\subseteq M(2,\mathbb{C})\cong\mathbb{R}^8, where M(2,\mathbb{C}) denotes the set of 2 by 2 complex matrices.

    When f is viewed as a function \widetilde{f}: \mathbb{R}^4\to\mathbb{R}^8, it is clear that \widetilde{f} and \widetilde{f}^{-1} are smooth maps since their component functions are of class C^\infty. Since SU(2) and S^3 are submanifolds, the restrictions to these submanifolds (i.e.\ f and f^{-1}) are also smooth.

    Hence f is a diffeomorphism.

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    Don’t fear the Monad

    Math Online Tom Circle

    Brian Beckman:

    You can understand Monad without too much Category Theory.

    Functional Programming = using functions to compose from small functions to very complex software (eg. Nuclear system, driverless car software…).

    Advantages of Functional Programming:

    • Strong Types Safety: detect bugs at compile time.
    • Data Protection thru Immutability: Share data safely in Concurrent / Parallel processing.
    • Software ‘Componentisation’ ie Modularity: Each function always returns the same result, ease of software reliability testing.

    Each “small” function is a Monoid.
    f : a -> a (from input of type ‘a‘ , returns type ‘a’)
    g: a -> a

    compose h from f & g : (strong TYPING !!)
    h = f。g : a -> a

    [Note]: Object in Category, usually called Type in Haskell, eg.’a’ = Integer)

    You already know a Monoid (or Category in general) : eg Clock

    1. Objects: 1 2 3 …12 (hours)
    2. Arrow

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    Programming and Math

    Math Online Tom Circle

    Category Theory (CT) is like Design Pattern, only difference is CT is a better mathematical pattern which you can prove, also it has no “SIDE-EFFECT” and with strongTyping.

    The examples use Haskell to explain the basic category theory : product, sum, isomorphism, fusion, cancellation, functor…

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    Echelon Form Lemma (Column Echelon vs Smith Normal Form)

    The pivots in column-echelon form are the same as the diagonal elements in (Smith) normal form. Moreover, the degree of the basis elements on pivot rows is the same in both forms.

    Proof:
    Due to the initial sort, the degree of row basis elements \hat{e}_i is monotonically decreasing from the top row down. For each fixed column j, \deg e_j is a constant. We have, \deg M_k(i,j)=\deg e_j-\deg \hat{e}_i. Hence, the degree of the elements in each column is monotonically increasing with row. That is, for fixed j, \deg M_k(i,j) is monotonically increasing as i increases.

    We may then eliminate non-zero elements below pivots using row operations that do not change the pivot elements or the degrees of the row basis elements. Finally, we place the matrix in (Smith) normal form with row and column swaps.

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    Persistent Homology Algorithm

    Algorithm for Fields
    In this section we describe an algorithm for computing persistent homology over a field.

    We use the small filtration as an example and compute over \mathbb{Z}_2, although the algorithm works for any field.
    A filtered simplicial complex with new simplices added at each stage. The integers on the bottom row corresponds to the degrees of the simplices of the filtration as homogenous elements of the persistence module.

    The persistence module corresponds to a \mathbb{Z}_2[t]-module by the correspondence in previous Theorem. In this section we use \{e_j\} and \{\hat{e}_i\} to denote homogeneous bases for C_k and C_{k-1} respectively.

    We have \partial_1(ab)=-t\cdot a+t\cdot b=t\cdot a+t\cdot b since we are computing over \mathbb{Z}_2. Then the representation matrix for \partial_1 is
    \displaystyle M_1=\begin{bmatrix}[c|ccccc]  &ab &bc &cd &ad &ac\\ \hline  d & 0 & 0 & t & t & 0\\  c & 0 & 1 & t & 0 & t^2\\  b & t & t & 0 & 0 & 0\\  a &t &0 &0 &t^2 &t^3  \end{bmatrix}.

    In general, any representation M_k of \partial_k has the following basic property: \displaystyle \deg\hat{e}_i+\deg M_k(i,j)=\deg e_j provided M_k(i,j)\neq 0.

    We need to represent \partial_k: C_k\to C_{k-1} relative to the standard basis for C_k and a homogenous basis for Z_{k-1}=\ker\partial_{k-1}. We then reduce the matrix according to the reduction algorithm described previously.

    We compute the representations inductively in dimension. Since \partial_0\equiv 0, Z_0=C_0 hence the standard basis may be used to represent \partial_1. Now, suppose we have a matrix representation M_k of \partial_k relative to the standard basis \{e_j\} for C_k and a homogeneous basis \{\hat{e}_i\} for Z_{k-1}.

    For the inductive step, we need to compute a homogeneous basis for Z_k and represent \partial_{k+1} relative to C_{k+1} and the homogeneous basis for Z_k. We first sort the basis \hat{e}_i in reverse degree order. Next, we make M_k into the column-echelon form \tilde{M}_k by Gaussian elimination on the columns, using elementary column operations. From linear algebra, we know that rank M_k=rank B_{k-1} is the number of pivots in the echelon form. The basis elements corresponding to non-pivot columns form the desired basis for Z_k.

    Source: “Computing Persistent Homology” by Zomorodian & Carlsson

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    BM Category Theory 10.1: Monads

    Math Online Tom Circle

    10.1Monads

    $latex begin{array}{|l|l|l|}
    hline
    Analogy & Compose & Identity
    hline
    Function & : : : : “.” & : : : : Id
    hline
    Monad & ” >> = ” (bind) & return :: “eta”
    hline
    end{array}$

    Imperative (with side effects eg. state, I/O, exception ) to Pure function by hiding or embellishment in Pure function but return “embellished” result.

    Monad = functor T + 2 natural transformations

    $latex boxed {text {Monad} = {T , eta , mu} }&fg=aa0000&s=3$

    $latex eta :: Id dotto T$
    $latex mu :: T^{2} dotto T$
    $latex text {Natural Transformation : } dotto $

    http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html#functors

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    Category Theory 9: Natural Transformations, BiCategories

    Math Online Tom Circle

    In essence, in all kinds of Math, we do 3 things:

    1) Find Pattern among objects (numbers, shapes, …),
    2) Operate inside the objects (+ – × / …),
    3) Swap the object without modifying it (rotate, flip, move around, exchange…).

    Category consists of :
    1) Find pattern thru Universal Construction in Objects (Set, Group, Ring, Vector Space, anything )
    2) Functor which operates on 1).
    3) Natural Transformation as in 3).

    $latex boxed {text {Natural Transformation}}&fg=000000&s=3$
    $latex Updownarrow $

    $latex boxed {text {Morphism of Functors}}&fg=aa0000&s=3$

    Analogy:

    Functors (F, G) :=operation inside a container
    $latex boxed { F :: X to F_{X}, : F :: Y to F_{Y}}&fg=0000aa&s=3$

    $latex boxed {G :: X to G_{X}, : G :: Y to G_{Y}}&fg=00aa00&s=3$

    Natural Transformation ($latex {eta_{X}, eta_{Y}}&fg=ee0000&s=3$) := swap the content ( $latex F_{X} text { with } G_{X}, F_{Y} text { with } G_{Y} $) in the…

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    French New Math Lichnerowicz Pedagogy 

    Math Online Tom Circle

    https://en.m.wikipedia.org/wiki/Andr%C3%A9_Lichnerowicz


    See the 1970s FrenchBaccalaureate Math Textbooks:(for UK Cambridge GCE A-level Math students, this is totally new “New Math” to us !)

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    Curry-Howard-Lambek Isomorphism

    Math Online Tom Circle

    Curry-Howard-Lambek Isomorphism:

    $latex boxed {text {Category Theory = High School Algebra = Logic = Lambda Calculus (IT)}}&fg=aa0000&s=3$

    Below the lecturer said every aspect of Math can be folded out from Category Theory, then why not start teaching Category Theory in school.

    That was the idea proposed by Alexander Grothendieckto the Bourbakian Mathematicians who rewrote all Math textbooks after WW2, instead of in Set Theory, should switch to Category Theory. His idea was turned down by André Weil.

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    Fighting spam with Haskell | Engineering Blog | Facebook Code

    Math Online Tom Circle

    Facebook rewrote the SPAM rule-based AI engine (“Sigma“) with Haskell functional programming to filter 1 million requests / second.

    The Myths about Haskell : Academia, Not for Production ?

    Why Facebook choosesHaskell Functional language for Spam rule engine?

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    Everyone’s Unique Timezone (Motivational)

    Relax. Take a deep breath. Don’t compare yourself with others. The world is full of all kinds of people – those who get successful early in life and those who do later. There are those who get married at 25 but divorced at 30, and there are also those who find love at 40, never to part with them again. Henry Ford was 45 when he designed his revolutionary Model T car. A simple WhatsApp forward message makes so much sense here:

    “You are unique, don’t compare yourself to others.

    Someone graduated at the age of 22, yet waited 5 years before securing a good job; and there is another who graduated at 27 and secured employment immediately!

    Someone became CEO at 25 and died at 50 while another became a CEO at 50 and lived to 90 years.

    Everyone works based on their ‘Time Zone’. People can have things worked out only according to their pace.

    Work in your “time zone”. Your Colleagues, friends, younger ones might “seem” to go ahead of you. May be some might “seem” behind you. Everyone is in this world running their own race on their own lane in their own time. God has a different plan for everybody. Time is the difference.

    Obama retires at 55, Trump resumes at 70. Don’t envy them or mock them, it’s their ‘Time Zone.’ You are in yours!” 

    Source: http://www.mensxp.com/special-features/today/34993-if-you-think-you-are-going-nowhere-in-life-take-a-deep-breath-and-read-this.html

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