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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…
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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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    De Rham Cohomology

    De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined.

    Definition:
    A differential form \omega on a manifold M is said to be closed if d\omega=0, and exact if \omega=d\tau for some \tau of degree one less.

    Corollary:
    Since d^2=0, every exact form is closed.

    Definition:
    Let Z^k(M) be the vector space of all closed k-forms on M.

    Let B^k(M) be the vector space of all exact k-forms on M.

    Since every exact form is closed, hence B^k(M)\subseteq Z^k(M).

    The de Rham cohomology of M in degree k is defined as the quotient vector space \displaystyle H^k(M):=Z^k(M)/B^k(M).

    The quotient vector space construction induces an equivalence relation on Z^k(M):

    w'\sim w in Z^k(M) iff w'-w\in B^k(M) iff w'=w+d\tau for some exact form d\tau.

    The equivalence class of a closed form \omega is called its cohomology class and denoted by [\omega].

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    Haskell Tutorial in One Video

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    Singular Homology

    A singular n-simplex in a space X is a map \sigma: \Delta^n\to X. Let C_n(X) be the free abelian group with basis the set of singular n-simplices in X. Elements of C_n(X), called singular n-chains, are finite formal sums \sum_i n_i\sigma_i for n_i\in\mathbb{Z} and \sigma_i: \Delta^n\to X. A boundary map \partial_n: C_n(X)\to C_{n-1}(X) is defined by \displaystyle \partial_n(\sigma)=\sum_i(-1)^i\sigma|[v_0,\dots,\widehat{v_i},\dots,v_n].

    The singular homology group is defined as H_n(X):=\text{Ker}\partial_n/\text{Im}\partial_{n+1}.

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    Mapping Cone Theorem

    Mapping cone
    Let f:(X,x_0)\to (Y,y_0) be a map in \mathscr{PT}. We construct the mapping cone Y\cup_f CX=Y\vee CX/\sim, where [1,x]\in CX is identified with f(x)\in Y for all x\in X.

    Proposition:
    For any map g: (Y,y_0)\to (Z,z_0) we have g\circ f\simeq z_0 if and only if g has an extension h: (Y\cup_f CX,*)\to(Z,z_0) to Y\cup_f CX.

    Proof:
    By an earlier proposition (2.32 in \cite{Switzer2002}), g\circ f\simeq z_0 iff g\circ f has an extension \psi: (CX,*)\to (Z,z_0).

    (\implies) If g\circ f\simeq z_0, define \tilde{h}: Y\vee CX\to Z by \tilde{h}(y_0,[t,x])=\psi[t,x], \tilde{h}(y,[0,x])=g(y). Note that \tilde{h}(y_0,[0,x])=\psi[0,x]=g(y_0)=z_0. Since \displaystyle \tilde{h}(y_0,[1,x])=\psi[1,x]=gf(x)=\tilde{h}(f(x),[0,x]), \tilde{h} induces a map h: Y\cup_f CX\to Z which satisfies h[y,[0,x]]=\tilde{h}(y,[0,x])=g(y). That is h|_Y=g.

    (\impliedby) If g has an extension h: (Y\cup_f CX,*)\to (Z,z_0), then define \psi: CX\to Z by \psi([t,x])=h[y_0,[t,x]]. We have \psi([0,x])=h[y_0,[0,x]]=z_0. Then \displaystyle \psi([1,x])=h[y_0,[1,x]]=h[f(x),[0,x]]=gf(x). That is, \psi|_X=g\circ f.

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    BM Category Theory II 1.1: Declarative vs Imperative Approach

    Math Online Tom Circle

    Excellent lecture using Physics and IT to illustrate the 2 totally different approaches in Programming:

    1. Imperative (or Procedural) – micro-steps or Local 微观世界
    2. Declarative (or Functional) – Macro-view or Global 大千世界

    In Math:

    1. Analysis (Calculus)
    2. Algebra (Structures, Category)

    In Physics:

    1. Newton (Law of Motions), Maxwell (equations)
    2. Fermat (*) (Light travels in least time), Feynman (Quantum Physics).

    In IT: Neural Network (AI) uses both 1 & 2.

    More examples…

    In Medicine:

    1. Western Medicine: germs/ viruses, anatomy, surgery
    2. Traditional Chinese Medicine (中医): Accupunture, Qi, Yin-Yang.

    Note (*): Fermat : My alma mater university in Toulouse (France) named after this 17CE amateur mathematician, who worked in day time as a Chief Judge, after works spending time in Math and Physcis. He co-invented Analytic Geometry (with Descartes), Probability (with Pascal), also was the “Father of Number Theory” (The Fermat’s ‘Little’ TheoremandThe Fermat’s ‘Last’ Theorem). He…

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    BM Category Theory 3.x Monoid, Kleisli Category (Monad)… Free Monoid 

    Math Online Tom Circle

    [Continued from 1.1 to 2.2]

    3.1 MonoidM (m, m)

    Same meaning in Category as in Set: Only 1 object, Associative, Identity

    Thin / Thick Category:

    • “Thin” with only 1 arrow between 2 objects;
    • “Thick” with many arrows between 2 objects.

    Arrow : relation between 2 objects. We don’t care what an arrow actually is (may be total / partial order relations like = or $latex leq $, or any relation), just treat arrow abstractly.

    Note: Category Theory’s “Abstract Nonsense” is like Buddhism “空即色, 色即空” (Form = Emptiness).

    Example ofMonoid: String Concatenation: identity = Null string.

    Strong Typing: function f calls function g, both types must match.

    Weak Typing: no need to match type. eg. Monoid.

    Category induces a Hom-Set: (Set of “Arrows”, aka Homomorphism同态, which preserves structure after the “Arrow”)

    • C (a, b) : a -> b
    • C (a,a) for Monoid…

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    QS University Ranking 2017 by Math Subject

    Math Online Tom Circle

    Top 5:

    1. MIT
    2. Harvard
    3. Stanford
    4. Oxford
    5. Cambridge

    18 University of Tokyo

    20 Peking University 北京大学

    22 Ecole Polytechnique (France)

    26 TsingHua University 清华大学

    28 Hong Kong University

    32 Ecole Normale Supérieure (Paris)


    https://www.theguardian.com/education/2017/mar/08/qs-world-university-rankings-2017-mathematics

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    记叙文开头的几种方式 How to Write The Start of Narrative Composition

    Chinese Tuition Singapore

    记叙文是以记人,叙事,写景,状物(描绘事物)为主,主要内容是人物的经历和事物的发展变化。

    记叙文有五种主要表达方式:叙述,描写,议论,抒情,说明。而记叙文的开头主要有以下几种形式:

    一,叙述

    把人物的经历和事物的发展变化过程表现出来。用简单的话说,就是,这件事怎么发生的,过程是什么,结果怎么样。当然,如果用这种方式开头,就不需要把整个事情的过程交代清楚,一般只要把事情的起因表述清楚即可。过程和结果可以在正文中体现。

    比如:

    1. 描写母爱——上幼儿园的时候,妈妈给我买了一把可爱的小花伞。伞的大小对我来说刚刚好,因为正好能遮住我小小的身体。妈妈说,自从我有了小花伞,就特别喜欢下雨天。只要一下雨,我就会把小花伞找出来,拉着妈妈往外跑。妈妈就撑起一把大伞来遮住我的小伞,陪着我在雨里玩。

    2. 描写一次难忘的经历——清晨,大街上异常忙碌,人来人往,像一条畅流的小溪。忽然,两辆自行车撞在一起,像一块石头横挡在小溪中间,小溪变得流动缓慢,渐渐停止了。

    3. 描写一次闯祸——在故事发生时,他还是个七八岁的孩子,他常常做些让大人们意想不到的恶作剧。但是,因为他还只是个孩子,所以大人们除了偶尔斥责几句之外,都不把他做的那些调皮捣蛋的事放在心上。就这样,他的胆子越来越大,闯的祸也越来越大。

    作文开头交代了事情的起因,下面就可以直接写事情的经过。
    二, 描写

    主要是对人物的外貌,动作,心理,事物的形态,样貌等具体的刻画。通常对人物的这种描写会从侧面反映出人物的性格特点。

    例如:

    1. 描写邻居——我有一位小邻居,她的名字叫小红,今年九岁。她远远的小脑袋上扎着两条小辫子,有着一双水灵灵的大眼睛。她的耳朵粉红小巧,像贝壳一样。红嘟嘟的小嘴整天叽叽喳喳不知疲倦。

    2. 描写亲人——我弟弟很可爱,他那圆圆的小脸蛋上嵌着一双水灵灵的大眼睛。嘴唇薄薄的,一笑小嘴一咧,眼睛一眯,还生出一堆小酒窝,非常可爱。要是谁惹他生气了,他就会瞪大眼睛,撅起小嘴。

    如果作文中需要写关于某个人的事情,那在作文的一开始就告诉读者这个人的性格特点,将会为作文的正文做好铺垫。
    三, 抒情

    通过文中要描写的人或事来表达自己内心的情感。

    例如:

    描写母爱——如果说我有向全世界的人宣布一件事情的权力的的话,我一定会说,我要感谢那个赋予我生命,教会我勇敢,关爱我成长的,我心中最漂亮的女人-妈妈。

    如果作文题目是关于“后悔”,“感激”,“难过”等对一个人或一件事的心情,在作文开头就表现出这种情感是一个很好的选择。
    四, 回忆

    通常用于写时间比较久的事情,比如,童年,几年前,几个月前,等发生的事情。

    例如:

    1. 描写童年——在偌大的世界上,人人都有一个栖息之地—家庭。有的家庭富丽堂皇,有的家庭美满甜蜜。对无忧无虑的小孩子来说,这是一块充满慈爱和乐趣的生命之地。然而,我是个不幸的孤儿,从小失去了父母,跟姐姐住在外婆家。回忆起自己在外婆家度过的那几年,我的泪水就像断了线的珠子。

    2. 描写父爱——在我的记忆中,爸爸的背是温暖的。
    五,开门见山

    这是最常见的一种开头。也是最直接的一种方式。

    例如:

    描写一次难忘的回忆——在我的人生中,有许许多多的人生第一次,令我终身难忘的是第一次游泳。
    六,悬念

    作文的开头通过描写事情发展中最精彩的部分,即人物的动作或者语言,来引起读者的兴趣。

    例如:

    “这样的事做不得!”看着背影远去的小明,我从心中发出一声呼唤。当时,我真的应该阻止他的。

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    Euler’s formula with introductory group theory

    Math Online Tom Circle

    During the 19th century French Revolution, a young French boyEvariste Galoisself-studied Math and invented a totally strange math called “Group Theory“, in his own saying – “A new Mathnot on calculation but on reasoning”. During his short tragic life (21 years) his work was not understood by the world masters like Cauchy, Fourier, Poisson, Gauss, Jacob…

    “Group Theory” is the foundation of Modern Math today.

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    Category Theory : Motivation and Philosophy

    Math Online Tom Circle

    Object-Oriented has 2 weaknesses for Concurrency and Parallel programming :

    1. Hidden Mutating States;
    2. Data Sharing.

    Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra… =>

    $latex boxed {text {CT reveals the way how our brain works by analysing, reasoning about structures
    !}}&fg=aa0000&s=3$

    Our brain works by: 1) Abstraction 2) Composition 3) Identity (to identify)

    What is a Category ?
    1) Abstraction:

    • Objects
    • Morphism (Arrow)

    2) Composition: Associative
    3) Identity

    Notes:

    • Small Category with “Set” as object.
    • Large Category without Set as object.
    • Morphism is a Set : “Hom” Set.


    Example in Programming
    :

    • Object : Types Set
    • Morphism : Function “Sin” converts degree to R: $latex sin frac {pi}{2} = 1$

    Note: We just look at the Category “Types Set” from external Macroview, “forget ” what it contains, we only know the “composition” (Arrows) between the Category “Type Set”, also “forget”…

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    A Category Language : Haskell

    Math Online Tom Circle

    Haskell is the purest Functional Language which is based on Category Theory.

    eBook:

    http://learnyouahaskell.com/chapters

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    Online Study Guide : Abstract Algebra

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    How to Take Great Notes – Study Tips – How to be a Great Student – Cornell Notes

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