December 31, 2016 Saturday

Bedtime Story 

Paradoxical Decomposition of Hyper-Oxford

This is what we will land up with after taking out the first redundant alphabet from each word of all the 26 volumes:

Volume A: A, AA, AAA, … , B, BA, BAA, … , C, … , Z, ZA, …

Volume B: A, AA, AAA, … , B, BA, BAA, … , C, … , Z, ZA, …

Volume C:  A, AA, AAA, … , B, BA, BAA, … , C, … , Z, ZA, …

Volume Z: A, AA, AAA, … , B, BA, BAA, … , C, … , Z, ZA, …

Voila! They are all identical!

Moreover, each one is identical to the original hyper-Oxford!

Isn’t this a paradox?

If you are seriously interested in this paradox, I would encourage you to read the book:

The Pea and the Sun: A Mathematical Paradox

By Leonard M. Wapner

In a way, what we did with the hyper-Oxford is paradoxical decomposition.

We broke up the first infinite set into a finite number of disjoint infinite subsets.   

   

The paradox is the fact that all the 26 subsets can be mapped back into the original set using only finitely many distinct functions (like taking off or adding on the first alphabet).

This is exactly how Felix Hausdorff defined paradoxical decomposition.

We are going to do something similar with the sphere.

We are going to convert our hypothetical numerical sphere into infinite points thus decomposing it into a hyper-Oxford dictionary.   

The first step for that is to designate name for all the infinite points on the sphere.

Non-mathematically it can be done by giving each point a label depending upon how they are reached from a given starting point on the surface of the sphere which will be origin for that case.

We shall deal with the naming of all the points on the sphere in the nights to come.

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

                              

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

December 30, 2016 Friday

Bedtime Story 

Breaking Down the Super-Oxford in 26 Volumes 

Now is the time to show you a miracle guys.

If you take out all the first letters A from all the “words” that began with it, you end up creating once again the super-Oxford dictionary!

You need to go back to the last night’s bedtime story and have a look at all the words once again.

It may not be so clear even now, so let me explain this to you in more detail and stepwise.

Let us once again analyze this super-Oxford dictionary.

It runs something like this.

A, AA, AAA…, AB, ABA, ABAA, … , AC, …, AZ, AZA, …

B, BA, BAA…, BB, BBA, BBAA, … , BC,…, BZ, BZA, …

C, CA, CAA…, CC, CBA, CBAA, … , CC,…, CZ, CZA, …

Z, ZA, ZAA…, ZB, ZBA, ZBAA, … , ZC,…, ZZ, ZZA, …

Now, to simplify things further and make the super-Oxford less of a burden to carry around, we can divide this massive tome into 26 volumes, one for each alphabet.

So now this massive tome has 26 volumes which go as follows:

Volume A: A, AA, AAA…, AB, ABA, ABAA, … , AC, …, AZ, AZA, …

Volume B: B, BA, BAA…, BB, BBA, BBAA, … , BC,…, BZ, BZA, …

Volume C: C, CA, CAA…, CC, CBA, CBAA, … , CC,…, CZ, CZA, …

Volume Z: Z, ZA, ZAA…, ZB, ZBA, ZBAA, … , ZC,…, ZZ, ZZA, …

There is yet another thing which can be done to reduce the girth of each volume without the loss of any information.

As you would have noticed, all the words say in volume C begin with C and that makes the first letter C redundant.

This fact holds good for all the volumes.

So it makes sense to take out the first letter from each volume since we would know the first letter of each word which is fixed by default.

What do you think we will get if we carry out this step?

We shall probably discover it by next 24 hours if I stay alive.  

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

                              

Good night mon ami and my fellow cousin ape.

         

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

December 29, 2016 Thursday

Bedtime Story 

The Super Oxford Dictionary Analogy

Before we deal with Banach Tarski theorem, let me give an analogy (to the mathematical sphere) that originally came from the mathematician and writer Ian Stewart.

It will require a little imagination, a very little.

Let us consider a theoretical Oxford dictionary that has all the possible combinations of all 26 letters so that it will even have “words” that do not make any sense such as aa, aaaa, aaab, bbbb, zzzz and so on.

We can call it the super-Oxford dictionary.

It is will something like this:

A, AA, AAA, AAAA, AAAAA, AAAAA…

AB, AAB, AAAB, AAAAB, AAAAAB…

BA, BAA, BAAA, BAAAA, BAAAAA…

B, BB, BBB, BBBB, BBBBB, BBBBB…

AC, ACA, ACAA, ACAAA, ACAAAA…

AZ, AZA, AZAA, AZAAAZ…

Z, ZA, ZAA, ZAAA, ZAAAA…

Z, ZZ, ZZZ, ZZZZ, ZZZZZ…

Of course, it will also have words like words, noun, verb, animal and so on as is found in most typical dictionaries.

It is also bound to have such absurd combinations like swordverb, nountail and so on since it is supposed to have all possible combinations of letters.

It is a humongous and a massive super-dictionary with every possible combination of letters in it. 

Now you know what I mean.

Now look at this set of letters very carefully for what I am going to propose next will stun you.

For the magic, you will have to wait.

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

                              

Good night mon ami and my fellow cousin ape.

         

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

December 28, 2016 Wednesday

Bedtime Story 

Banach Tarski Theorem in Other Forms

Reassembly of all the infinite points of the sphere after its decomposition is again mathematical that involves rotation.

It is at this crucial point of reassembly that the axiom of choice is invoked to allow reconstruction of the sphere from an infinite and uncountable number of choices.

Let me list out all the other forms in which Banach Tarski Theorem can be stated:

2^nd Statement

A sphere in a Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent one, and reassembly.

3^rd Statement

A 3-D Euclidean sphere is equidecomposable with 2 copies of itself.

4^th Statement

Any two bounded subsets of a 3-D Euclidean space with non-empty interiors are equidecomposable.

5^th Statement

S² is SO(3)-paradoxical as is any sphere centered at the origin.

Moreover, any solid ball in R³ is G₃-paradoxical and R³ itself is paradoxical.

G₃ is the group of isometries in R³.

The group of isometries is the group of bijections from R³ to R³ that preserves distance.

The word paradox is also mathematical and needs to be precisely defined.

Consider a group G acting on a set X. 

So suppose there is a subset of set X that can be broken up into 2 disjoint sets each of which can be split up into finitely into many disjoint joints and then reassembled into the copy of the original.

This is paradoxical.

Enough of this mathematical jargon.

Let us now try to understand the chain of reasoning of the Banach Tarski Paradox in unpretentious simple English.

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

                              

Good night mon ami and my fellow cousin ape.

         

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

December 27, 2016 Tuesday

Bedtime Story 

Hausdorff Misses the Boat to America

As Anti-Semitism swept over Germany, at the University of Greifswald one of Hausdorff’s lectures was stopped by the Nazi students.

He was moving to and forth from Greifswald to the University of Bonn.

Over there he was forced to terminate his winter semester Calculus Program of 1934/35 academic year.

His professional academic career was brought to an end on March 31, 1935 by the university giving him the emeritus status without any acknowledgements of his everlasting contributions to mathematics.

Yet he continued to work on set theory and topology on his own, his friend borrowing and bringing him books from the library.

Like so many others, in 1939 he tried for a research scholarship to the land of free and plenty but he failed.

Only a fraction of European Jews such as Tarski, Gödel, von Neumann and Einstein were lucky enough to reach the land of rich and plenty and flourish there. 

After all, there is a limit to how many brains American universities can assimilate, then as it is now.

Then the orders came.

Somewhere in 1942, Felix Hausdorff along with his wife and his wife’s sister were ordered to head to the Endenich camp in Bonn which was essentially a transit camp before the culminating journey to the Final Solution.

Hausdorff knew that by missing the ticket to America his fate had been sealed.

In a joint pact, Hausdorff along with wife and sister-in-law consumed the first commercially available barbiturate sodium diethyl barbiturate on January 26, 1942.

In excess of course and died with dignity.

I personally would have done something very similar in order to avoid and go through the humiliation that was being meted out to the inmates of Auschwitz.  

All his handwritten manuscripts, notes that constitutes his Nachlass is now preserved in the University and State Library of Bonn, the least the nation could do as a mark of apology for the crimes committed by the Weimar state against its minorities.    

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

                              

Good night mon ami and my fellow cousin ape.

         

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

December 26, 2016 Monday

Bedtime Story 

Demystifying Banach Tarski Theorem

Based on all the above material, Banach and Tarski wrote that famous paper of theirs in 1924.

The paper itself does not label the work as paradox but merely as a theorem.

It is the writers of popular mathematics (is that an oxymoron, popular mathematics?) literature that love to baffle the gentry with the word paradox.

The Banach Tarski theorem or paradox can be stated in several ways and let me list a few of them out:

1^st statement

Given a solid ball in 3-D space, there exists a decomposition of the sphere into a finite number of disjoint subsets, which can then be reassembled back in a disparate way to furnish 2 identical copies of the authentic sphere.

I have to make a few points clear about the nature of the sphere, the act of decomposition and then the act of reassembly.

First of all, the sphere is more of a mathematical idea (rather than a physical atomic ball) which consist of arrangement of infinite points, the key word being infinite.

That means we are dealing with an infinite set which takes us back to the ideas of Cantor and Vitali set.

A sphere of radius 1 is mathematically defined as:

S = {(x, y, z)|x² + y² + z² <= 1}

The act of decomposition of the ball is yet again a mathematical one that was defined by Hausdorff which involves mathematical rotation and translation.

It is denoted by SO(3) which is a group of rotation about the origin of 3-D Euclidean space R3 under the operation of composition.

Hausdorff had proved that a surface of a unit sphere (radius 1) in space is a disjoint union of 3 sets B, C and D and a countable set E such that, on the one hand, B, C and D are pairwise congruent, and, on the other hand, B is congruent with the union of C and D.

This proof of Hausdorff is extremely fascinating and should itself be seen as a paradox by itself.

By the way, Hausdorff was the son of a Jewish merchant, and like so many others he was subjected to bitter humiliation after that infamous Kristallnacht of November 9, 1938.

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

                              

Good night mon ami and my fellow cousin ape.

         

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

December 25, 2016 Sunday

Bedtime Story 

Lebesgue Measure and Equidecomposability

Based on the Lebesgue measure, the Vitali Theorem can also be stated as follows:

There exist subsets of R (real numbers) that are not Lebesgue measurable.

Hence the Axiom of Choice needs to be invoked for dealing with Vitali sets.

One of the many interesting properties that ensues from Lebesgue measure is this:

If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and (A) is equal to the sum (or infinite series) of the measures of the involved measurable sets.

The implications of this is very counterintuitive.

It two open subsets of a line or a plane are equidecomposable then they have equal area.

This means that any infinitely sized sets when split into subsets will result in more infinitely sized subsets.

Let me give you an example of equidecomposability.

Take a unit circle A and another unit circle B with one missing point Z.

These two circles are equidecomposable simply because the second circle B can be made into a full complete circle.

Let me show you how.

In the circle B with the missing point Z make a subset E that consists of all the points that are positive integer number of radians clockwise from the missing point Z.

They are countable but infinite because of the irrationality of pie.

Let all the rest all the numbers on the circle B be defined under the subset F.

Then one pick up the subset E and rotate it anti clockwise by 1 radian.

The missing point at Z gets filled up by the point one radian to the right of it, just like the point on the n-1 radian will get filled by the point of the n radian.

Then put the subset E back with the subset F and you end up with a full circle B that is same as the full circle A.

This proves the unit circle A and the unit circle B with a missing point Z were and are equidecomposable.  

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

                              

Good night mon ami and my fellow cousin ape.

         

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

Felix Hausdorff had originally come with the concept of paradoxical decomposition of a set using the Axiom of Infinity. 

Paradoxical decomposition is partitioning of a set into 2 subsets along with an appropriate group of functions on some universe, such that the partition can be mapped back onto the entire original set using only finitely many distinct functions.    

December 24, 2016 Saturday

Bedtime Story 

Giuseppe Vitali, Felix Hausdorff and Henri Lebesgue had Laid the Groundwork

This paper “On the Decomposition of Point Sets into Respectively Congruent Parts” in turn was based on two papers published earlier, one by Giuseppe Vitali and the other by Felix Hausdorff.

Giuseppe Vitali in 1905 had described a set of real numbers that is not Lebesgue measurable, meaning even a unit set of real numbers, that is [0, 1] on a number line which is a subset of real numbers R has the same cardinality as that of the continuum R.   

 

This Vitali Set is based on the Axiom of Choice which allows selection of non-measurable infinite sets.

Felix Hausdorff went a step further and wrote a paper on the paradoxical decomposition of a sphere S² (a 2-dimensional sphere in R³) somewhere around that time.

It states that if a certain countable subset is removed from S², then the remainder can be divided into 3 disjoint subsets A, B and C such that each is congruent to the others and all are congruent to the set B  C.

Essentially what Hausdorff proved was that for a sphere S² there is no finitely additive measure that is defined an all subsets.

This implies that the measure of subset A is both 1/3 and ½ of the non-zero measure of the whole sphere S².

This concept, as we will see, becomes central to the paper published by Banach and Tarski.

Here, the mathematical idea of 3D rotation group plays an important role.

Mathematically, a rotation about an origin is defined a transformation that preserves the origin, Euclidean distance and orientation (handedness of space). 

A rotation group (group of all rotations about the origin of 3-D Euclidean space R³) is designated by SO(3).

French mathematician Henri Lebesgue made an important contribution to the set theory by introducing a novel concept to measure subsets of a set.

It goes by the name of Lebesgue measure and it assigns a measure to the subsets of n-dimensional Euclidean space.

For n = 1, 2 and 3 the Lebesgue measure happens to correspond with the standard measure of length, area and volume.

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

                              

Good night mon ami and my fellow cousin ape.

         

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

December 23, 2016 Friday

Bedtime Story

“On the Decomposition of Point Sets into Respectively Congruent Parts” by Banach and Tarski (1924)

It was not that Tarski wanted to leave Poland; for just as he was set to leave for the conference a vacancy had been created with the demise of Leśniewski that would have gone to him.

Inadvertently, his wife and children along with his entire extended family were left behind as the invasion of Poland was sudden and unexpected.

Almost his entire extended family was murdered and wiped out of existence by the efficient German killing machinery.

For all the personal loss that befell upon Tarski, there was one huge yield for him in the professional sense.

Fate was delivering him to the most impregnable and apt place where his talent would be recognized and nurtured.

Almost immediately on arriving into America, Tarski took up a position at Harvard and later in 1942, thanks to a grant through the Guggenheim Fellowship, he got a chance to be at the holiest grail of mathematics: the Institute of Advanced Study at Princeton.

That too at the time when it was studded with luminaries that had set the world of science on fire.

Over there, once again he encountered the great logician Kurt Gödel (At the end of each day, Einstein would walk back with Gödel to their respective homes).

Soon after his short stay at the IAS, he got a position at the Mathematics Department of University of California, Berkley which he held on till the very end.

Tarski met with phenomenal success in America in almost every way a mathematician can hope to achieve, being acknowledged as a great and passionate teacher, a charismatic speaker and publishing masterly papers in mathematics, logic and formalized languages.

He went on to be elected to the United States National Academy of Sciences, the British Academy and even the Royal Netherlands Academy of Arts and Sciences.

Yet, for all his success and achievements in America, the work for which he is most renowned came in his earliest days in his homeland.

It was his first paper on set theory that he wrote with Stephan Banach (University of Lwów) in 1924 when in Warsaw, Poland.

The paper was published in the journal Fundamenta Mathematicae and was titled:

“On the Decomposition of Point Sets into Respectively Congruent Parts”.

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

                              

Good night and my fellow cousin ape.

         

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

December 22, 2016 Thursday

Bedtime Story 

The Western Allies Jilt Poland - A Tragedy of Devastating Proportions

Both United Kingdom and France were trying to woo both Hitler and Stalin in an attempt to stop the carnage that would be unleashed if both these army were set loose over Europe.

In 1938, in order to appease Hitler, the British Prime Minister Neville Chamberlain betrayed Czechoslovakia (by agreeing to the annexation of Sudetenland), deeply upsetting and marginalizing Soviet Union and signed the Munich Agreement.

United Kingdom was willing to sacrifice all principles and all allies if it would avert war with the Weimar Republic.

Hitler was a master at signing agreements and renegading on them.

After seeing the rift between France and United Kingdom against Soviet Union, Hitler rushed his foreign minister to Moscow where on August 23, 1939 the Nazi-Soviet Pact was signed.

Just a week after signing the pact, on September 1, 1939 Germany invaded Poland.

Then a mere 17 days later after Soviet Union ended its hostilities with Japan, the Red Army invaded Poland from the east.

Hitler was not so sure about dealing with Czechoslovakia earlier but with Poland he was dead sure:

“The object of the war is…physically to destroy the enemy.

This is why I have prepared, for the moment only in the East, my ‘Death’s Head’ formations with orders to kill without pity or mercy all men, women and children of Polish descent or language.

Only in this way can we obtain the living space we need”.

This was Hitler’s roar to the military commanders at Obersalzberg just a week before the invasion of Poland.

That was the end of the short freedom that Poland had enjoyed in the brief interbellum. 

Poland was clearly abandoned, betrayed and sacrificed at the altar of inaction by its Western allies.

The cost of this mistake would be beyond any redemption.

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

                              

Good night and my fellow cousin ape.

         

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

September 01, 1939 German tanks Invading Poland