“Algebraic Topology” can detect the Multi-dimensional neural network in our brain – by studying the Homology (同调) and co-Homology (上同调) with the help of Linear Algebra (multi-dim Matrix) & Computers.
Homology = compute the number of “holes” in multi-dim space.
Neurons formed in the brain can be modeled in Math (Topology) by Simplex 单纯 (plural : Simplices), billions of them interconnected into a complex -“Simplicial Complex” (单纯复体)。
What is Math ?
https://infinityplusonemath.wordpress.com/2017/06/17/what-is-math/
Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…
What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years afterNewton & Leibnizhad invented it – “Epsilon-Delta” Analysis.
Difference between Riemann Integral & Lebesgue Integral ?
Professor Yitang Zhang is a famous Math professor who made important progress in number theory (Twin Prime Conjecture). Most strikingly, he made this progress in his fifties, which is kind of rare in the mathematical world.
Source: Quanta Magazine
Yitang Zhang on the beach adjoining the University of California, Santa Barbara, after scratching a function in the sand related to his current work on the Landau-Siegel zeros problem.
As an adolescent during the Cultural Revolution in China, Yitang Zhang wasn’t allowed to attend high school. Later, in his 30s, he worked odd jobs in the United States and sometimes slept in his car. But Zhang always believed he would solve a great math problem someday. Still, despite becoming one of China’s top math students and completing his doctorate at Purdue University in Indiana, for seven years Zhang could not find work as a mathematician. At one point, he worked at a friend’s Subway sandwich restaurant to pay the bills.
“I was not lucky,” Zhang, who is both incredibly reserved and self-confident, told Quanta in a 2015 interview.
At 44, after finally being hired to teach math at the University of New Hampshire, he turned his attention to number theory, a subject he had loved since childhood. He analyzed problems in his head during long walks near his home and the university. In his 50s, well past what many mathematicians consider their prime years (indeed, the Fields Medal is awarded to mathematicians under the age of 40), he began trying to prove the twin primes conjecture, which predicts an infinite number of prime number pairs that have a difference of two, such as 5 and 7, 29 and 31, and 191 and 193. No one had been able to prove this in over 150 years, and top number theorists could not even prove the existence of a bounded prime gap of any finite size.
In 2013, at 58, Zhang published his proof of a bounded prime gap below 70 million in one of the world’s most prestigious journals, the Annals of Mathematics. The paper’s referees wrote that Zhang, who had been unknown to established mathematicians, had proved “a landmark theorem in the distribution of prime numbers.”
Read more at: Quanta
Thanks to all readers of mathtuition88.com, I have received WordAds payout by WordPress.
Mathtuition88.com first joined WordAds in 01-2014, where the payout was quite miserable (less than $1 for 5000 views). The payout has since increased considerably, thanks to higher quality adverts localized in Singapore.
The advertisements shown are quite relevant (usually tuition by top tutors/schools, travel and other deals in Singapore), so do turn off ad-blockers when visiting mathtuition88.com. Thanks once again!
Singapore scores the 2017 PISA Test World’s First in Math.
Below is the “Singapore Math” shown by an USA School adopting the pedagogy.
I had the opportunity to meet the inventor of the “Singapore Math” Prof Lee PengYee (李彝义) in a Math seminar at NUS few years ago, who said he took the inspiration of the ancient Chinese math 算术 (literary means “Counting Technique” sans Algebra), combined with the PolyaProblem Solving Methodology, with a visual tool called “The Model Diagram”.
Note that this pedagogy is based on concrete visualisation, while good for majority of young students to learn computation and problem solving, it doesn’t train the opposite skill in “abstractness” which is required in university Advanced Math, where the French, Russian and USA university students excel more than the Asian cohorts.
The Ideal Math Education for the next 22ndCentury:
Singapore Math (Computation for…
View original post 19 more words
Step 1)
Visit https://appliedtopology.github.io/javaplex/ and download the Persistent Homology and Topological Data Analysis Library
2)
Download the tutorial at http://www.math.colostate.edu/~adams/research/javaplex_tutorial.pdf and jump to section 1.3. Installation for Matlab.
3)
In Matlab, change Matlab’s “Current Folder” to the directory matlab examples that you just extracted from the zip file.
(See https://www.mathworks.com/help/matlab/ref/cd.html to change current folder)
Type this in Matlab: cd /…/matlab_examples
Where … depends on where you put the folder
4) In the tutorial (from the link given in step 2), proceed to follow the instructions starting from “In Matlab, change Matlab’s “Current Folder” to the directory matlab examples that you just extracted from the zip file. In the Matlab command window, run the load javaplex.m file.”.
5) Test: Run example 3.2 (House example) by typing in the code (following the tutorial)
On Wikipedia (https://en.wikipedia.org/wiki/Dedekind_number), it is stated that the Dedekind’s number M(n) is the the number of abstract simplicial complexes with n elements.
This is incorrect, at least based on the Wikipedia definition of abstract simplicial complex, which does not allow the empty set as a face.
The correct definition is found in another Wikpedia site: https://en.wikipedia.org/wiki/Abstract_simplicial_complex
The number of abstract simplicial complexes on up to n elements is one less than the nth Dedekind number.
Habitica is a productivity app, something like a to-do list in a gamified form. Users gain experience and gold from completing tasks, and can join in quests with a party to defeat monsters.
The mage class has one of the highest damage in the game (after warrior), and its base stat is intelligence. The best mage gear with highest intelligence (as of June 2017) is as follows:
Armor: Jean Chalard’s Noble Tunic (+25 INT, +25 CON)
Headgear: Nameless Helm (+25 INT, +25 STR)
Shield-Hand Item: Diamond Stave (+16 INT)
Weapon: Trident of Crashing Tides (+15 INT)
For stats, put all stats into intelligence.
What is 17×6? (without using calculator)
Practicing basic mental math without calculator is good. Even for PSLE, where calculator is allowed, it is a good idea not to rely on it too much.
Three mysterious Math Objects:
are hiding in circle.
5 Math Tips That Will Speed Up Your Calculations
For most students, regardless of their age, studying math seems like a nightmare. Taking tuition for maths helps when you or your child struggle with calculations.
Check out these 5 simple tricks for making mental math faster and more accurate:
If the total is more than 9, add 1 to the first digit and insert the second number, e.g. 78 x 11 = 7(7+8)8, or 7(15)8. Move the 1, so 7+1(5)8 = 858.
What about 8% of 350? Add up the 8s for each 100, and half of an 8 for the remaining 50, so (8 x 3) + 4 = 28. The same for 8 x 35, but moving the decimal point, so 2.8. Percentages can also be flipped, so 32% of 5 is the same as 5% of 32.
Math doesn’t have to be scary, and an online math tutor can help you deal with advanced problems without getting overwhelmed.
About Author: Making education simple and easy to comprehend is Dana Jandhayala’s forte. She’s had a long career as an educator where she has taught in several different schools and institutes in multiple countries. Today, she helps students with personalized online tutorials by SchoolPage that help make concepts easy to understand, making learning fast and fun. She writes to help students study better, and to coach parents so they can facilitate the success of their children.
After the defeat in 1st match, Ke Jie said “AlphaGo is approaching God!”.
This version of AlphaGo is 10x more powerful than last year’s version which has beateb the South-korean Go player Lee.
https://www.theguardian.com/technology/2017/may/23/alphago-google-ai-beats-ke-jie-china-go
Kotlin is the “New Java” officially supported by Google from May 2017! It is less verbose (罗唆) than Java which is clumsy with boilerplates (样板), interoperates with Java on JVM, with modern functional programming features, and most importantly,it is Multi-Platform : Java, Android, Javascript, and future versions run as native codes on iOS, MacOS and Linux (Microsoft – work in progress). This eliminates the current headache of having to re-write the same applications for different platforms in different languages.
Google makes Kotlin a first-class language for writing Android apps
https://blog.plan99.net/kotlin-fp-3bf63a17d64a
Kotlin Tutorials:
https://kotlinlang.org/docs/tutorials/
https://developer.android.com/kotlin/get-started.html
Hands-on:
https://try.kotlinlang.org/#/Kotlin%20Koans/Introduction/Hello,%20world!/Task.kt
Two ways to program in Kotlin:
1) Google way: Download Android Studio 3.0 (with Kotlin and Java 8 Support):
2) (A better way): use Jetbrains “Intelli IDEA for kotlin” (bundled with Kotlin)
https://kotlinlang.org/docs/tutorials/getting-started.html
Books:
https://antonioleiva.com/kotlin-android-developers-book/
https://www.amazon.com/gp/aw/s/ref=is_s_ss_i_5_6?k=kotlin+for+android+developers&sprefix=kotlin
Just read this very nice article on Quora, on the relationship between Math and Chess: https://www.quora.com/What-math-tricks-are-hidden-in-chess
Also interesting is this YouTube documentary “My Brilliant Brain” featuring Susan Polgar.
There’s lots of clever math tricks hidden in chess. Here are some examples.
Pawn race: Want to know if you can catch your opponents pawn before it becomes a queen? Use simple geometry! Draw a diagonal line starting from the pawn to the first row, then make a box. If your king can enter that box, he can catch the pawn! No need to count moves or calculate.
Opposition: Want to know if you have the distant opposition in a king and pawn engame? Well you could calculate it out, or you could use simple parity! If the king’s are on the same line (diagonally, horizontally, or vertically) look at how many squares are in-between the kings. If it’s your turn: an even number of squares means you have the opposition, an odd number of squares means your opponent has the opposition. There’s six squares between the kings in the diagram below. If it’s white’s turn, they have the opposition!
Calculating the relative value of pieces:
If you randomly place a piece on an empty chess board, how many moves will it have on average? I calculated this out a long time ago, then I compared it to what the pieces are assumed to be worth.
This is over-simplistic, but interesting. So it looks like the king has a relative value of 4, and Bishop’s are slightly better than knights. The downside of only looking at available moves is that it ignores things like the fact that the Bishop can only attack one color square, or how pieces work together. The Bishop pair, for example, is worth more than the sum of each individual bishop, as they complement each other. This fact is used in chess engine programming.
Shortcut to calculating knight moves:
I use this trick all the time in my games. Want to know how long it takes a knight to get somewhere? Simply memorize this square. The knight square is a beautifully symmetrical pattern, which makes it really easy to memorize. Each number represents the number of moves it will take the knight to reach that square.
Interesting fact – If you wanna reach any of the fours, then any move the knight makes will get you closer to reaching that square.
Knight’s also operate according to parity. That is, they attack dark squares, then light squares, then dark squares, and so on. See the pattern above – even moves get you to the same color square that your sitting on, odd moves get you to the opposite colored square.
Pythagorean Theorem does not apply in chess:
Our brains are hardwired to think that diagonals are longer than edges of a square. This is not the case in chess. This can lead to some interesting positions. For example:
White to move. It looks like black is going to promote their pawn and white will lose. If white tries to promote their pawn right away the black king will catch it. Miraculously, white can actually force a draw here. They do so by marching their king diagonally as shown. They will either catch the black pawn or promote their white pawn. This is only possible because it takes the same number of moves for white to move their king from h8 – e5 t- h2 as it does to go in a straight line from h8 to h2. Counter intuitively, going diagonally doesn’t increase the trip length.
Centralization matters! Except for rooks…
(Almost) Every piece in chess will have more available moves in the center of the board than they will on the edge of the board. More available moves = more control. This is why chess strategy tells you to always control the center.
This doesn’t apply to rooks… Like the honey badger, rooks just don’t care. If you place a rook on an empty board, they will always have 14 moves. Doesn’t matter where you place them. This is due to the square symmetry of the board, which lines up with the way rooks move.
The 8 Queens Puzzle:
How many Queens can you place on a chessboard such that none of them could capture each other? 8 of course. After all, a chess board is just an 8 by 8 grid. But trying to find the solution is much more difficult. There are actually 96 solutions (12 if you don’t count symmetry or rotations). This is a complex math and computer science problem. They have solved the same puzzle for every solution on a 27 by 27 sized board.
Counting the number of possible games in chess:
A clever mathematician named Claude Shannon calculated the upper bound limit for chess, resulting in about 10^120 possible different games. This was a huge influence on the field of computer science and chess, and shows why we cannot solve chess using brute force.
There are many other interesting math tricks hidden in chess. I look forward to reading about others.
Emmanuel Macron is the youngest French President (39) since Napoleon Bonaparte (40).
A brilliant student since young, he impressed his secondary school Drama teacher 24 years older, finally married her.
Like any genius (Einstein, Galois, Edison, …) who doesn’t adapt well in the traditional education system, Macron entered the prestigious and highly competitiveClasse Préparatoire (Art Stream) Lycée Henri IVin Paris to prepare for the “Concours” (法国抄袭自中国的)”科举” Entrance Examsin France’s top Ecole Normale Supérieure (ENS). Like the 19CE Math geniusEvariste Galoiswho failed the Ecole Polytechnique Concours twice in 2 consecutive years, Macron also failed ENS “Concours” in 2 consecutive years.
He revealed recently, ”The truth was I didn’t play the game. I was too much in love (with my former teacher) for seriously preparing the Concours …”
Note: French traditional name for the elitist tertiary education (first 2 or 3 years if repeat…
View original post 104 more words
This site (https://www.sharelatex.com/blog/2013/08/09/thesis-series-pt5.html) has a guide on how to create your own 
The actual Tex source code is found here: https://www.sharelatex.com/project/51fa85c3db89c3c351085071.
I like it as the source code is neat and clean, you can easily edit it if you know some basic Latex. Some other templates out there are quite complex and convoluted, it is hard to customize it.
Tutor: Mr Wu (Raffles Alumni, NUS Maths Grad)
Email: mathtuition88@gmail.com
Syllabus: Primary / Secondary Maths Olympiad. Includes Number Theory, Geometry, Combinatorics, Sequences, Series, and more. Flexible curriculum tailored to student’s needs. I can provide material, or teach from any preferred material that the student has.
Target audience: For students with strong interest in Maths. Suitable for those preparing for Olympiad competitions, DSA, GEP, or just learning for personal interest.
Location: West / Central Singapore at student’s home
Very good idea by NYGH. Sleep is important for students.
SINGAPORE: For almost a year now, Nanyang Girls High (NYGH) students have been starting school at 8.15am – a good 45 minutes later than most secondary schools.
And the results have been telling.
The school in Bukit Timah has been taking part in ground-breaking sleep studies conducted by Duke-NUS Medical School researchers – whose studies have shown that 80 per cent of teens here don’t get enough sleep, which affects their health, grades and cognitive abilities.
It was what the teachers of NYGH had been suspecting all along.
Mrs Ho-Sam Choon Juen, NYGH dean of student systems and info management, said: “For a long time, we’d known that our girls were not sleeping enough because of their academic and extra-curricular demands.
Read more at http://www.channelnewsasia.com/news/cnainsider/a-big-difference-in-students-after-nanyang-girls-starts-school-8833844
Maths Tuition
Tutor (Mr Wu):
– Raffles Alumni
– NUS 1st Class Honours in Mathematics
Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Also has experience teaching Additional Math (O Level, IP).
Personality: Friendly, patient and good at explaining complicated concepts in a simple manner. Provides tips for how to check for careless mistakes, and tackle challenging problems.
Email: mathtuition88@gmail.com
Areas teaching (West / Central Singapore, including Bukit Batok, Dover, Clementi, Jurong)
[Continued from previous BM Category Theory …]
$latex boxed {
text {type Algebra f a = f a} to text {a}
}&fg=aa0000&s=3$
Intuition: [Artificial Intelligence] You teach the computer, like a Primary 6 kid, that Algebra is atype of expression (f) which, after evaluation, returns a value.
If a = i (initial)[or u (terminal)],
$latex boxed {
text {(f i} to text {i )} implies
text {f = Fix-point}
}&fg=0000aa&s=3$
Intuition: Fix-point because, the Initial “i” after the evaluating the expression f, returns itself “i”.
Lambek’s Lemma
$latex boxed {
text {Endo-functor = Isomormphism}
}&fg=00aa00&s=3$
Note: Endo-functor is a functor (equivalent tofunction in Set Theory) within the same Category (Endo = Self = 自)
Video 8.1F-Algebras & Lambek’s Lemma
Video 8.2Catamorphism & Anamorphism
foldr ~ catamorphism (浅层变质) of a Fix-point endo-functor on a List.
Examples: Fibinacci, Sum_List
View original post 147 more words
Maths Tuition
Tutor (Mr Wu):
– Raffles Alumni
– NUS 1st Class Honours in Mathematics
Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Also has experience teaching Additional Math (O Level, IP).
Personality: Friendly, patient and good at explaining complicated concepts in a simple manner. Provides tips for how to check for careless mistakes, and tackle challenging problems.
Email: mathtuition88@gmail.com
Areas teaching (West / Central Singapore, including Bukit Batok, Dover, Clementi, Jurong)
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.
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!
Nice traditional hymn.
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.
Abstract Algebra studies all kinds ofmathematical structures(Group, Ring, Vector Space, Category, …) , and relationship between them if the structures are preserved after mapping.
Recently I discovered a classical guitar Christian composer on YouTube – John Michael Talbot. His music is quite amazing, and one of a kind.
2. Be not afraid
Corrinne May is a Singaporean singer.
Set 
When we pass to the derived couple, each group 
The maps 
![j_2(i_1a)=[j_1a]](https://s0.wp.com/latex.php?latex=j_2%28i_1a%29%3D%5Bj_1a%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
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.
Given a space 
We have a chain complex 
Relative cycles
Elements of 
Relative boundary
A relative cycle 
Long Exact Sequence (Relative Homology)
There is a long exact sequence of homology groups:
The boundary map 
![[\alpha]\in H_n(X,A)](https://s0.wp.com/latex.php?latex=%5B%5Calpha%5D%5Cin+H_n%28X%2CA%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![\partial[\alpha]](https://s0.wp.com/latex.php?latex=%5Cpartial%5B%5Calpha%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
Source: http://www.aquinascollege.edu/prayer-before-study-exams-spring-2013/
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.
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.
Exact sequence (Quotient space)
If 
where 
Reduced homology of spheres (Proof)
For 
Exactness of the sequence then implies that the maps 
Define the reduced homology groups 
Relation between 
Since 
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?
[Hint]:
$latex (a + x)^{frac {1}{3}} + (a – x)^{frac {1}{3}} = 2(a)^{frac {1}{3}} + 2(x)^{frac {1}{3}} $
Terence Tao:
3 Phases of Math Training:
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:
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
Mayer-Vietoris Sequence
For a pair of subspaces 
Example:
Let 
We obtain isomorphisms 
Wishing all readers a happy Easter!
This is a famous Spanish Easter Song: Resucitó, with translation.
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.
Amazing organ music.
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.
Quite a nice video here:
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.
Mathtuition88 