October 31, 2017 Tuesday

Bedtime Story 

The Fathers of Modern Microbiology 

The moment from replicating molecules to cells with walls was a landmark moment in the history of life on our planet as it allowed the invisible hand of natural selection to alter the game completely.  

This essentially meant that now if any “lucky” molecule by random process generated an enzyme that had better survival role then selection pressure would be able to “work” on it.

The person who actually categorized the Archaea as a whole separate domain and kingdom of life was an American microbiologist and biophysicist Carl Woese, essentially an east coast guy.

Even before him, the guys who first divided all cellular organizations into prokaryotes and eukaryotes were Roger Stanier and C. B. van Niel.

Stanier was Canadian and van Niel as the name suggests was from Holland, all of them migrating to America, that had become the power house of pure science after the World War II.

It was in their highly influential 1962 paper that they defined prokaryotes as those organisms that lacked a cell nucleus.

If Stanier and van Niel divided all current organisms existing into earth into two domains of prokaryotes and eukaryotes, then it Carl Woese gave origin to the third domain, the Archaea.

The domain was founded essentially on the basis of genetic phylogeny, more specifically on 16s ribosomal RNA.

The gene for the 16s ribosomal RNA is an extremely reliable molecular clock as it is a highly conserved gene among bacteria and Archaea.        

So these simplest humble creatures were the subject of study of an equally humble unknown microbiologist Francisco Mojica from an equally unknown University of Alicante.

From 1993 onwards, Mojica published several papers based on his research.

Let me name a few of them for you.

I know many would prefer not to know the details how science is done, but I think it is important to have an understanding how scientific research is really done.

It is not glamorous as made out either in short BBC-type documentaries nor sensational as newspapers and tabloid headings make it appear to be.

It is slow, tedious yet meticulous, often-boring repeated work with test tubes, centrifuges, culture plates, incubators, spectrometers, stirrers, shakers and so on.

Each experiment finally ends with collection of date that gets fed into a software program for analysis followed by discussion of the findings and possible conclusion with the mentors.

Following the completion of all the experiments, the conclusion is written down in the form of scientific paper that is proof read by all the co-authors and then finally the principal scientist.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 30, 2017 Monday

Bedtime Story 

Archaea - Our Most Ancient Cousins

As I keep saying, most of even very prominent scientists die generally as unknown figures.

In fact, it can be almost guaranteed that if I were to ask you to name any living contemporary scientist in any field, may it be astrophysics or mathematics or molecular biology or microbiology or even any field of your choice you or your memory would be very hard pressed to recall a name. 

To recall any microbiologist would be perhaps next to impossible for microbiology after all is one of the driest of all the subjects, especially for medical students.

I still recall how we would dread to memorize the names of various pathological microbes and their culture characteristics, culture in this case not very different from the way we would define culture for human apes.

Yet even I as a medical student never recall having been taught about a group of organisms called Archaea.

Till date I had never come across Haloferax and Haloarcula, single celled organisms with just glycerol-ether lipids for cell membrane, a circular chromosome with no nucleus or any membrane bound organelle?

Amoeba yes, but Haloferax and Haloarcula…No way!

Their metabolism is quite strange and varied, some using sunlight, some using energy trapped in inorganic chemical bonds and some using organic compounds to obtain energy.

Even those that use sunlight are not obligatory photosynthetic, but have devised mechanisms of electron transport chains or direct proton pumping to establish electro-chemical gradient.

This is all that the enzyme ATP synthase needs to provide molecular energy currency for these archaic cells.

Worse of all they never have sex and are total celibate!

They are so primitive that they have been around here since the birth of life and hence we apes gave them the heading of archaea which in Greek means “ancient things”.

It has to be stressed that archaea are distinct from bacteria and that they are no more related to each other than each is related to eukaryotes.

All these three groups shared ancestral colony of organisms but they later on diverged in three different ways.

Much before cells evolved some 3.8 billion years ago when there existed just the replicator molecules, lateral gene transfer was very convenient and almost unrestricted, and      Darwinian natural selection was unable to exist its pressure.

It meant that even an “unlucky” replicator molecule would get to share the benefits of a “lucky” replicator and thus genetic advantage was pooled.

But such free lateral transfer of gene probably allowed fixation of certain subsets of genes.

This would become important later when the basic lipid membrane formed around the replicating molecules, isolating the lucky from the unlucky replicators.

Formation of this lipid barrier around the replicators is called the Darwinian transition by the evolutionary biologists.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 29, 2017 Sunday

Bedtime Story 

Mysterious Palindromic Interrupted Clustered Repeats

Yoshizumi Ishino at the Osaka University was primarily interested in the cloning of gene iap of E. coli that coded for the enzyme alkaline phosphatase.

This is an enzyme that removes the phosphate group from organic compounds by hydrolysis and is widely found in the cells of organisms, both eukaryotes and prokaryotes.

On cloning the iap gene, Ishino got the expected iap product that contained 345 amino acids.

But along with that he also unexpectedly got some mysterious short, palindromic repeat sequences of nucleotides.

(I must add that a genetic palindrome of nucleotides is slightly different from that which is conventionally understood in its English meaning and I shall not go further than that).

Ishino had no idea what these interrupted clustered repeats were, so he simply stated that in his paper and moved on.

As a matter of fact he moved on to the laboratory of Dieter Söll at Yale who had in his turn been tutored under the great scientist Har Gobind Khorana at the University of Wisconsin-Madison.    

Nothing much was reported on these mysterious clustered repeats for the next six years.

Subsequent progress that happened in this area came unexpectedly from two papers originating from Netherlands.

Both the papers were published in the year 1993, one in August and the other in September, the former in the Journal of Clinical Microbiology and the later in Molecular Biology.

The first paper was titled “Comparison of various repetitive DNA elements as genetic markers for strain differentiation and epidemiology of Mycobacterium tuberculosis”.

The second paper was titled “Nature of DNA polymorphism in the direct repeat cluster of Mycobacterium tuberculosis; application for strain differentiation by a novel typing method.”

As you can see from the titles of both the two papers, the Netherland team of molecular biologists never used the term CRISPR but simply used the terms cluster and repeats.

Moreover, they seemed not to have given much importance to the findings and all that they could think of was to exploit the diversity of these cluster repeats of DNA in M. tuberculosis in classifying this clinically important microbe into different strains.        

So these cluster repeats of DNA was then primarily being considered as a taxonomical tool for the various strains of M. tuberculosis.

At about the same time in the early 1990s, a microbiologist by the name of Francisco Mojica was investigating two organisms from the kingdom Archaea at the University of Alicante in Spain.

As you will yourself agree, all the three nouns of interest in the above sentence are totally nondescript; the microbiologist, the university and the organisms.

Yet all will play a significant role in our story that will continue to play out in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 28, 2017 Saturday

Bedtime Story 

Learning to Accept the Futility 

Just look at the letter that was sent to Dawkins by an academic from New Zealand:

“When Lenin traveled through Germany earlier this century (twentieth which means the last century), the Germans permitted him only to travel in a sealed, locked train – on the condition that he proceeded nonstop from the one border post to the other.

They clearly recognized his persuasiveness and power of his ideas and their capacity to produce unhappiness.

I respectfully request you that you don’t lend Dawkins’s book to anybody for the same reason.”

In short, Dawkins’s ideas and views are not only radical but the reality is utterly pessimistic in the sense that there is no higher purpose that human apes aspire for or there is no after life as they wish for.

As Steven Weinberg puts it, “The more the universe seems comprehensible, the more it also seems pointless.”

That is the brutal reality and we must accept it and if sounds pessimistic, then so be it.

As grown up adults with reason and intellect, one must learn to accept and handle not only the tragedies of personal lives but also the futility or rather the pointlessness of our existence.  

Dawkins is reviled by a large swath of population for his forthrightness but I on the other hand revere Richard Dawkins for his direct and crystal clear writing and plain-speaking documentaries.

Even my hero Isaac Asimov pales in comparison, when it comes to candidness and unadulterated expression of scientific views that calls for total annihilation of religious and superstitious world view.

Thanks to the ideas of men like Richard Dawkins, now we have fields such as computational biology that feed on using genetic algorithms to imitate the effects of evolution on silicon chips, just much faster.   

At this point, I would like to slightly digress from my primary story and tell you something about CRISPR/Cas system.

The reason for digressing to the CRISPR/Cas system is to show you in complete detail how utterly and comprehensively digital life is.

It will require a patient hearing and I would like to start from the scratch.

This story begins in the Osaka University of Japan.          

 

Osaka University is one of Japan’s Seven National Universities that ranks as the fourth best higher educational institute of Japan.

In 1987 a molecular biologist by the name of Yoshizumi Ishino published a paper in the Journal of Bacteriology with four other co-authors.

The paper was titled “Nucleotide sequence of the iap gene, responsible for alkaline phosphatase isozyme conversion in Escherichia coli, and identification of the gene product”.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 27, 2017 Friday

Bedtime Story 

Craig Venter and Richard Dawkins on Life being Digital Information

The closer the geneticists and computer scientists look at life, the more they find that it resembles information technology.

If there remains any doubt, it should go away simply by the fact how the length of human genome is described.

Human genome is described in base pairs and is over 3 billion base pair long where the base refers to DNA nucleotides.

Here at this point I can’t help myself quoting some sentences from the book “Life at the Speed of Light” by Craig Venter that is to some extent based on the lecture that he delivered at the Trinity College, Dublin on July 12, 2012 – some seven decades after Erwin Schrödinger had given his classical oration on life.

“Over the next sixty minutes I explained how life ultimately consists of DNA-driven biological machines.

   

All living cells run on DNA software, which directs hundreds to thousands of protein robots.

We have been digitizing life for decades, since we first figured out how to read the software of life by sequencing DNA.

Now we can go in the other direction by starting with the computerized digital code, designing a new form of life, chemically synthesizing it DNA, and then booting it up to produce the actual organism.

And because the information is now digital we can send it anywhere at the speed of light and re-create the DNA and life at the other end.

In the span of a single lifetime, we have advanced from Schrödinger’s “aperiodic crystal” to an understanding of genetic code to the proof, through construction of a synthetic chromosome and hence a synthetic cell, that DNA is the software of life.”

It is a book that was strongly recommended by Mon Ami to me, and hence not only does it lie in my collection of book shelf but it has a special attribute of having been read twice by ours truly.

If Craig Venter does not impress you that much, then I would recommend you my hero Richard Dawkins.

Any book of his will do but here in our case we shall take the recourse to The Blind Watchmaker.

   

The evolutionary biologist in his 1986 book – The Blind Watchmaker (a must read!) wrote this following lines:

“What lies at the heart of every living thing is not a fire, warm breath nor ‘a spark of life’.

It is information, words, instructions…

Think of a billion discrete digital characters…

If you want to understand life think about digital technology.”

To many, Dawkins is controversial, spiteful, venomous and even dangerous.

I will read out just one of the benign hate mail that he received in the nights to come to give a demonstration of how much he is reviled in some circles (and I think he sedretly enjoys it as it is a vital contributory factor to his “popularity”).

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 26, 2017 Thursday

Bedtime Story 

Y Combinator has a Biological Analog

I have found it interesting that most computer scientists and very smart men such as Alan Turing, Erwin Schrödinger, Claude Shannon and John von Neumann among others have found strong association between living and non-living, between humans and machines and between computers and brain.

Once they saw that if out of mathematics can arise mathematical logic, and electrical circuits arranged in specific ways can make intelligent decisions, then surely from the far more enormous complexities of cellular machinery and neurons can far easily arise life and consciousness.  

If Erwin Schrödinger defined life in terms of thermodynamics then John von Neumann invented the concept of cellular automata.

Alan Turing though widely known for his Turing Machine and cryptanalysis was perhaps one of the first mathematical biologists who published his masterpiece “The Chemical Basis of Morphogenesis” in 1952 at the age of 39.

With his two-part article published in 1948, “A Mathematical Theory of Communication”, Claude Shannon introduced the information theory that is the converging intersection of man fields ranging from mathematics to biology, from computer science to neurobiology and from pattern recognition to natural language processing.   

Like all these great men listed above, someone else noticed a connection between the loop function/Y combinator of Haskell Curry and a fundamental element of biology.  

Have a look at them once again carefully.

rec = λf. (λx. f (x x)) (λx. f (x x))

loop = (λx.x x) (λx.x x)

In both of them we have two functions lying next to each other and both these either enable loop or recursion.

Now where in biology can you imagine something very similar happening at a very fundamental level?

If you are not able to get it, let me give you a hint.

Look at the sequence below.

(5’->3’) ATGGAATTCTCGCTC (coding strand)

(3’->5’) TACCTTAAGAGCGAG (template strand)

Now you will get it!

It is the DNA, the code of life!

DNA is double helix structure with two coils of information lying side by side which enables them to replicate with high fidelity.

The Y combinator in an eerily similar manner has two pieces of information lying side by side that enables recursion not only in lambda calculus but all functional programming languages.

Self replication and recursion in a way are very similar process and this may not be a mere coincidence.

May be I can say as a conjecture that at a very fundamental lever there could be a connection between pure abstract mathematics and information theory and biology.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 25, 2017 Wednesday

Bedtime Story 

λf. (λx. f (x x)) (λx. f (x x))

Last night I had left you with one possible definition of recursion in lambda calculus:

rec = λf. (λx. f (x x)) (λx. f (x x))

This specific definition of recursion function in lambda calculus is known as the Y combinator which was the eventual target from the very beginning.

This was discovered by Haskell Curry and I have no idea why he or anybody else gave this function the name of Y combinator.

This definition of recursion very understandably looks to be daunting and intimidating, but I beg you to place it next to our initial definition of loop and compare them.

rec = λf. (λx. f (x x)) (λx. f (x x))

loop = (λx.x x) (λx.x x)

Do you see that uncanny resemblance?

In both the formulations, there are exact two functions placed next to each other.

In the original looping case the function is (λx.x x) and in the case of Y combinator it is (λx. f (x x).

In both the instances, the idea of self application is being applied.

The only significant difference is that in the case of Y combinator there exists that additional function f within the function.

That f is needed to make the Y combinator fit the definition of recursion as recursion demands that a function be defined in its own terms as we saw in the case of factorials.

This Y combinator encodes recursion into lambda calculus.

It may look benign and simple but it is a powerful idea, an idea as powerful as elementary logic gates or in biology that cells are the fundamental units of structure and function in all living organisms.

Strangely enough, that last sentence is one of the three tenets that go under the rubric of cell theory.

The word “theory” very often leads to unnecessary debates and proclaimed by deniers of fact with great gusto as “only a theory”.

Evolution of life is heatedly debated in the same manner which only demonstrates lack of knowledge on that subject.  

Anyway, let us not get into silly debates and return back to the Y combinator and its power to encode recursion into lambda calculus.    

This simple idea is deployed in all functional programming languages in computer science.

I think I am almost done with Y combinator, buy hey wait a second!

There is something very interesting about Y combinator that even many computer science guys probably will find novel and noteworthy and that I shall take up in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 24, 2017 Tuesday

Bedtime Story 

Encoding Recursion

The function ‘loop = (λx.x x) (λx.x x)’ is unique.

It contains a self application function within which is another self application function.

This function is sufficient to perform looping or recursion.

We can verify it if it is really so.

Let us look at this function more carefully (λx.x x) (λx.x x)

  

What will the function λx do if we feed to it the input x?

It will take in input x and give the output x x placing the two x side by side.

In our larger case the input is not x but (λx.x x)

So in this instance (λx.x x) will be substituted for x in the first part of (λx.x x).

Doing so will generate back (λx.x x) (λx.x x)

You can carry this process on and on and it will only return back the same value.

This was a very simple example of a loop in lambda calculus.

Let us see if we can broaden our view a bit and generalize the idea of recursion.

Let us define recursive function as follows:

rec f = f (rec f)

Here is a function that will take recursive function as input.

Note mon ami how this one differs from the previous example that we had considered.

The previous example was Loop = Loop and then we had applied lambda calculus to it.

So all it gave was same thing again and again going around in loops.

But our current example is not a loop but a recursion as there is a function f sitting between two rec f.

On “running” this function, we will get this:

f (f (f (f (f…))))

This function will go on infinitely forever.

This is the basis or the idea behind general recursion in computer science.

If this function can be embedded into the lambda calculus, then recursion will emerge in it.

One possible definition of recursion in lambda calculus is this:

rec = λf. (λx. f (x x)) (λx. f (x x))

I will not go into its derivation as that may not make for an engaging bedtime story for everyone though mon ami will love it.

We shall work on it in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

October 23, 2017 Monday

Bedtime story 

Decision Making and Loops

Last night we were recapitulating the basics of lambda calculus and recalling how we had constructed the true and false function in it.

True = λx.λy.x

False = λx.λy.y  

The True function takes in two variables and gives as output the first variable whereas the False function takes in two variables and gives as output the second variable.

With these two function then, one can go on to define other higher logical functions or gates such as NOT, AND or OR.

These True and False functions are very critical to programming as they are fundamental to decision making and problem solving activities.

With just these two simple lambda calculus expressions, True = λx.λy.x and False = λx.λy.y , you already have a powerful mechanism to make decisions given two choices.

     

Decision making and problem solving is not only required by simple machines such as air conditioners or a refrigerators or calculators but even living organisms.

The next question that we have to deal with is how to encode recursion into lambda calculus using such simple expressions.

One way is to simplify things and reduce the problem.

Let us see what can be the simplest definition for a recursive function.

Well, the simplest recursion would be a program that loops.

Just look at the following single line program.

Loop = Loop

It hardly needs any explanation that this program will continue to go round and round in loops in a single place.

It is perhaps the simplest recursive program that can be written down.

Now can this simplest loop be encoded into the lambda calculus?

Perhaps yes and the technique that can be used is self application.  

In self application, a function is applied to itself.

Let us first define our loop.

loop = (λx.x x) (λx.x x)

We have placed copies of two functions side by side.

In lambda calculus placing a function side by side means a function is being applied to itself.

Now look at each function itself.

(λx.x x) implies that the function λx takes in x and applies it to x.

So this function too is a self application.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg