Happy New Year 2021

 Naya saal mubarak ho, hai meri pooja kai;

Yahi samay hai karne ka, nahin dooja koi.

Goodbye 2020

Goodbye 2020. Today is 31st December 2020.

Here's wishing a very happy 2021 with this expression.

Once again wishing all my blog readers a very happy new year 2021. I hope your wishes come true in the new year.

This is important

 

The expression 2n/((2n-1)(2n+1)) works both with three consecutive numbers and four consecutive numbers. Only thing is with three consecutive numbers it is equal with sigma for 1 to infinity. And with four consecutive numbers it is equal with integral for 1 to infinity.

Here are Wolfram Alpha screenshots

An equation in the last week of 2020

 Here is an equation in the last week of 2020. A similar equation was made earlier. I have added another variable m to what I did earlier.

Kindly check the equation for random values of m and n. Revert to me back if there is an error.

Have a nice week, dear readers !!

A Big Statement

A big statement from a man of limited learning. Forgive me if I am wrong. Consider me suitably if I am right.

I have extended what I had done earlier. See this and this.

Corrected expression as on 2021/03/04 is below:-

Planned journey of a self guru

Firstly, I hope the meeting between farmers and government meets success. The meeting is scheduled on Tuesday the December 29th. I urge both the government and farmers to focus on success of the meeting. That is both should prepare to give in.

Secondly, I wish my mathematical ambition gets a lift. My e, beta function and gamma function are strong. I have also given explanations on Fourier series and partial differential equations in my blog. So heat equations should be an achievable target. SAIL officials if are reading this, I request to give me a pull.

Thirdly, art drawing. I have made plans for 6 months. First three months I shall work on perspective and shading. After finishing to considerable improvement; I shall work on hands and folds in clothes. The next three months.

Three important thoughts

First thought

When we think of a number; we feel it should not be 0 or infinity. In other words G and 1/G should be valid statements. Another thing is Pythagoras law should be valid. In other words square root 2 and square root of cos(x) and sin(x) should be valid.

Gauss constant says that.

Second thought

The philosophy of pluses and minuses is true. In other words philosophy of the mathematical constant pi is true.

Third thought

Man indulges in counting 1,2,3 . . . However the mathematical constant e is true.

I have explained the philosophies of pi, e, G and earlier gamma (Euler - Mascheroni constant).

I hope you like these philosophies. In case you feel different, then please add comments. I will be waiting for your comments.

Till then have a great day. Hope you have a enjoyable week ahead. And a great new year after that.

An intuition proved right

About 4 years back a thought came in my mind. 1 to 2 is hundred percent rise. But 99 to 100 is just about one percent rise. Is there any "constant" for this I wondered.

I will explain it. Only today it became clearer.

Consider 1/1, 1/2, 1/3 and so on. It decreases considerably.

Now consider 1/2,2/3,3/4 and so on. It increases but not considerably.

If we take to the power of 1/2,2/3,3/4 and so on then the increase may be considerable.

So the "constant" I wrote in first paragraph is 1/n +log(n)-log(n+1) summed up for n from 1 to infinity. Isn't it?

Well that is what is Euler-Mascheroni constant. Gamma. 0.577. . . . .

How is this for beauty ? My master Euler it was. Amazingly beautiful !!

The Golden Ratio

Remember the post on strange things in infinite series. In case you have missed, here is the link.

Now, take a look at the image below.

This one is the third one on Golden Ratio.

Before this one there is one on Pentagon construction. And another one on agm(7,8,10)/agm(1,2,3,4).

And today is Indian Mathematics day. What a great thing !!

Today (22nd December) is Indian Mathematics Day

Today is Indian Mathematics Day. The birth anniversary of Srinivasa Ramanujan. He was born on December 22, 1887 in Erode, Tamil Nadu.

His major contributions include Ramanujan Tau Function, Ramanujan Theta Function. About these two I have written blog posts. He also has done work on continued fractions. And on nested radicals. And more. Basically these are the areas I am aware of. I think there are hundreds of areas he has touched and left mathematicians puzzled.

On this eve, I would like to present an expression of Pi. I do not think I have presented an exact equation like this earlier in my blog posts. 

Here is a Wolfram snapshot too.

Have a great day blog readers!!

The beautiful four

This idea came into my mind and I tried the expressions in Wolfram Alpha. I do not think I tried this idea earlier.

Many people who do not understand what I talk about I want to tell this.

The general expression which I try out is ((mn+m-1)/(mn+m) - (mn-1)/(mn)). Putting m equal to 1 gives n/(n+1)-(n-1)/n; three consecutive numbers. Putting m equal to 2 gives (2n+1)/(2n+2)-(2n-1)/(2n); four consecutive numbers.

My whole thing I had been focusing on for the last one year or so is summed up in above. This idea of ((mn+m-1)/(mn+m) - (mn-1)/(mn)) has been tried on finite sigma, infinite sigma, binomial, integral, complex function, Gauss etc. And it has fetched clear cut results. This implies ((mn+m-1)/(mn+m) - (mn-1)/(mn)) is a great function. I have published papers in JETIR and IJMTT.

Right now as I have written in the first paragraph, I tried my ideas more. And here is the result.

Why should I do it? Revelation of numbers in hour glass is the reason. I believe revelation by nature/God.

Have a nice day, dear readers !!

On RCEP and farm laws

 Kya kimat doge janaab gehoon ka ya jwaar ka;

Apni kimat ka pata nahin kya kahoon kuch aur ka.

Indian farmers have a mentality of above and they are right. It is difficult to bring farming under "Indian economy" as we know. 

Take my case. I am definitely good in mathematics. And what have these people in government done. Nothing. And "Indian economy" they talk of. You need to think of profitable moves dear. Not partisan moves. K K Birla used to say "Take care of small things ". How right he was !!

Instead of making India move towards RCEP the government is taking on its own farmers. And it is a pity. Farmers are loosing lives.

Open up the banking. Make it 100 percent transparent. We will then move towards RCEP. Then apply for membership. This is what India needs to do. And not farmers.

This is interesting

 

The above is so interesting.

If you take the sigma, the negative 1 term and n factorial common on the left hand side and add; basically n(2n -1) + n is 2n*n. Of course you cannot do this on infinite sigma. But the result on LHS is 2nsquared.

And the result on RHS is nsquared.

There is one on Ramanujan too. 1+2+3+4+ ........ upto infinity is -1/12. If you don't believe this, see Youtube.

Strange ways of infinity. Infinity is closer than you think. Just rotate about yourself and you are talking of infinity. And take a scale (however long) you are talking of finite.

Analysis of Infinite by Euler in two parts must be a great book. But then it is very costly and is unaffordable in general standard.

Have a great day, dear blog readers !!

Four expressions

 The below four expressions are one and the same.

1. The expression on the leftmost side are 4 numbers. (mn) and (mn-1); and (mn+m) and (mn+m-1).

2. The expression on second from left are numbers in arithmetic-geometric  progression.

3. The expression on third from left is factorials.

4. The expression on right is reciprocals/

Do you like it? Or do you dislike these type of posts? Do let me know.

If you dislike, I shall discontinue posting on these type of posts.

A new expression

I was thinking on an idea. My idea when expressed in terms of n seemed to be n/((n+1)^2) alternately negated and added.

I used Wolfram Alpha to know what the expression fetches.

Here is the result

And here is the Wolfram screenshot

Hope you like the expression.

Have a nice day dear blog readers.

Prime numbers generator

Euler, my master made a terrific prime number generator years ago. The formula was m*m-m+41.

I thought m*m+m+1 was a great prime number generator. This was two years back when I wrote the paper on "Rational Number Series". Then I followed Euler somewhat and made it m*m+m+41. Also I tried m*m+m+2657.

With m*m+m+1, 311 prime numbers are generated for m equal to 1 to 1000.

With m*m+m+41, 580 prime numbers are generated for m equal to 1 to 1000.

With m*m+m+2657, 498 prime numbers are generated for m equal to 1 to 1000.

Here are the snapshots of the results.

Special thanks to onlinegdb.com for the C compiler.

My point in writing this article is this. Euler was so great. It was 17th century. Still he managed to get the magical number of 41. 

If you are curious how 2657; well it is nearest odd integer to 16.3 into 163. 163 being a favorite number of Ramanujan.

I think we can go ahead and try with nearest odd integer to 1.1 into 11, 1.9 into 19, 4.3 into 43 or 6.7 into 67.  Please try these if you are stuck up wondering what to do during COVID.

Have a nice day, dear blog readers. And in case you can beat by generating more prime numbers; do let me know.

In drawing, I intend to . . .

 In drawing, I intend to do the following in the coming days :-

1. Use the grey paper I have for shading practices. Professional shading is possible in these grey paper as the papers are soft.

2. Use white pencil too for highlighting.

3. Use my mobile camera and photo editor for black and white photos. These photos shall be the subject for my shading exercises on grey paper.  I shall take photos of fruits, utensils, vegetables etc. at home and practice.

The above is one part. The other part is normal white A-4 sheets.

4. In normal A-4 sheets, I intend to practice Asaro head at various angles. This will help me in perspective and proportions part of my portrait drawing journey.

5. In normal A-4 sheets, I intend to practice individual face parts like nose, eye etc. too.

So, all the necessary steps that are needed for drawing from a photograph can be practiced if I sincerely follow the above 5 things.

Drawing Practices

 

I do practices like above.

Tremendous improvement is needed to reach "professional" level. My level is that of "non-professional".

In the above:-

1. Ear anatomy is wrong.

2. Smoky lines used in all. Even in the landscape sketch.

I also play chess in tab. I am Level 6 to Level 7.

In mathematics, I am wondering now whether (mn+m-1)/(mn+m) - (mn-1)/(mn) is natural or not. n/(n+1) definitely is. I would have made a function of (mn+m-1)/(mn+m) - (mn-1)/(mn).(Please refer below @@)

Have a nice day, dear blog readers !!

@@

In Carnot cycle we have this equation:-

v2/v3 = v1/v2. This is a sort of n/(n+1) - (n-1)/n. Therefore, my argument for a function for (mn+m-1)/(mn+m) - (mn-1)/(mn) is valid. As putting m =1 in (mn+m-1)/(mn+m) - (mn-1)/(mn) result in n/(n+1) - (n-1)/n.

I did SWOT diagram . It was similar to Carnot cycle somewhat. Strength is opposite to Weakness. Opportunity is opposite to Threats. And if Strength is high and Opportunity is high a man acts. And if Weakness is high and Threats are large a man does not act. A sort of Carnot cycle. So again n/(n+1) - (n-1)/n is valid. And therefore a function of the type (mn+m-1)/(mn+m) - (mn-1)/(mn).

Hope I have made my point clear. My point was advocating (mn+m-1)/(mn+m) - (mn-1)/(mn).

I want to make this clear

My caricatures began when I wanted to know how the musicians of RDBurman looked like.

Then I bought a book of Andrew Loomis on drawing heads. I realised I lacked 3d.

Meanwhile I did cartoons. Because I felt I had good humour.

If you take portraits, there are so many ways. Oil, watercolours, pencils. Even in pencils Andrew Loomis has one style while William Maughan has another.

I shall be only using black pencils on white or sometimes grey paper. In grey paper I shall use white pencil too. 

I wish to follow only da vinci. Sfumato technique for drawing. Smoky lines.

Here is a screenshot of Da vinci portrait of a head. Drawn in 1503.

I wish to achieve perfection like above drawing. I don't intend to do portraits like most artists do from a photo. Many of them are damn too good. No doubt.

But I shall only follow da vinci type of techniques. And in pencils. 

Landscape sketching with pencil

I shall be doing landscape sketches using pencils. This I shall be doing because I have reached 80 percent of my journey in portraits from photographs. I need to get out of my rut of just doing portraits from photographs.

I have bought  a kindle book on landscape drawing with pencil by Frank M Rines.

The technique for pencil art (landscape drawing) shall be smoky lines. The technique that I use for portraits from photographs. I don't intend to change this technique.

Many people may criticise for not using the standard shading techniques like cross-hatching. I think I can use smoky lines for shading. I have looked into internet.

I shall post images of my landscape drawings later. Till then, good day wishes dear readers.

On Self-guru-ing

 

Self-guru-ing is not an easy task. It is difficult.It requires mind control and a great sense of logic. The rewards that it will fetch is that you get to learn an art or learn a management paradigm. This may give you extra money too.

The most important thing in self-guru-ing is not having desires. That is you do not indulge in learning the art or the management paradigm on desires. You must have a concrete reason in your mind. However the willingness to learn the art or the management paradigm is a wish. Not a desire.

There are certain things you must be clear in your mind about the art or the management paradigm. As an example let us consider drawing. Whether pencil or color. If color, whether water-color or oil. If pencil whether portraits or cartoons. If you feel you want to learn all then it will be difficult. Though it may be wish and not a desire. Still it will be tough. Clarity and focus are essential.This calls for practicality of ideas.

When you start working on your art, you will realize that there are many aspects to the art that you are self-guru-ing. As an example, when you do portraits, you will realize that proportions, perspective, shading are so important. My point is that you must perfect one of these first. And only then should you proceed to the other two. Example, you concentrate on proportions and improve it to considerable level. Then you concentrate on shading. Otherwise the errors will all sum-up and you will end up in confusion.

De-motivation happens in self-guru-ing. Guruji will help you in overcoming de-motivation and you will certainly move ahead. Have faith in your abilities. Look out to internet for solutions. Apply thinking to the problem you face. Gradually your de-motivating factor will vanish and you will get back your confidence.

Khuda haafiz !!

Taylor series and Rational number series

There is a very compelling and interesting similarity in Taylor series and Rational number series.

It is indeed a great honor that Rational number series has a similarity with Taylor series.

Please refer to below material :-

The topmost statement is the Taylor series.

The second statement is Rational number series expression.

The third is deduced from the second expression.

The fourth has been modified from the third expression based on a mathematical expression I made and posted earlier.

Additions to tea in India

In India, Assam tea is the most preferred tea. And the most consumed. 

Normally one teaspoon of tea from the packet is added to boiling water. It is allowed to simmer in the hot water for about a minute. Then sugar and milk is added. This mixture is allowed to simmer in medium flame for about a minute. The tea is now ready.

Many times people add additives to tea to beat stress, soothe coughs, enhance mood and add flavor.

Some of the additives I have tried are:-

1. Tea Masala - Tea masala is a ready-made product available in market. It is manufactured by large spice making companies like Everest etc. Half a teaspoon for a cup of tea is ideal to enhance the flavor of tea.Tea that we have at about 11 am or so are best enjoyed if tea masala is added to it.

2. Himalayan Amrit Herbal tea - This product is meant to be taken alone as a herbal tea. But what I do with Himalayan Amrit Herbal Tea is that I add half a teaspoon of it to the regular Assam tea being prepared. This way not only do I manage to control cost but I also get the flavour of the Himalayan Amrit herbal tea. This is best enjoyed at about 5 pm with music switched on.

3. Ginger, cardamom, cloves, cinnamon, black pepper etc. - Depending on the mood and your perception of your health the above can be ground and put to the regular Assam tea being prepared.

4. Readymade tea with Mulethi - Nowadays teas with mulethi and other herbs are readily available as a packet. This tea is good for health and we do not have to add anything more to it. This has to be prepared like a regular tea.

That's all folks for now. Have a very nice day, dear blog readers.

No errors aimed for

 I am aiming for no errors. There may be over performance or under performance. But zero errors aimed for.

Gulshan Grover. White sheet. HB pencil.

Presently

 

The drawings are okayish. 

The problem is one does not come to know where one is wrong. If alone. I ask my mother and am posting the images after she is able to recognise the faces.

Another thing is I use images in mobile. Neither a laptop or a printout.

In case you fail to recognise the above are drawings of Amitabh Bachhan and Kamalhaasan.

Goodbye dear readers. Have a wonderful day ahead.

Self-guru-ing my way ahead

Self-guru-ing is a bad thing. It is dreadful. But in India you cannot help it. Because the gurus may not be appropriate. Anyway it is a very big discussion as to who is the right guru. Guru Nanak of course is. And it is Sikhism I look forward to when I write this article. 

I drew this. Grey paper used. White pencil used. 4B pencil used. No proportional divider used.

It is Jeevan. The bollywood character actor. Usually villian in bollywood movies.

My point is that to improve I have to self-guru my way. 

Because in India there is no way a commoner can learn to draw shadows of a white statue.

Have a great day, dear readers.

Shading face is working

 

Shading faces is working for me. It seems to be the case.

I was good in shading clothes. An idea cropped up. What if the face is covered with cloth? What if the cloth becomes thinner and stickier? Then I began to work. This idea is working for me.

You may try with the idea written in previous paragraph. And tell me if it works for you.

Good day wishes, dear blog reader.

Heads and clothes

 

The above is what I have been doing. Drawing faces in 3d and with experimentation. And shading clothes.

What is not happening ?

1. Shading faces

2. Portraits from photographs

3. Drawing the actual Asaro head

I don't think I can ever do the above 3 things. I have accepted that. 

Have a great day dear blog readers.

My efforts

I am working on 3d look.

I am practising and the faces look 3d now. I follow a mixture of Andrew Loomis and Asaro head principles. I draw four parallel arcs. And a hemisphere suitably. Remember the hemisphere has to be sliced suitably. Then I begin from ear. From ear arcs to nose line, chin line and two lines to eye. Then lip ends to eye balls which is almost straight. Then from nose balls to lower side chin. Lower side chin to nose-eye joint. Then upper side chin to side eye brow. 

Practising above gets me planes of faces. Then I work on planes of head. Then eyes as much as possible. Then nose planes as much as possible. 

My faces are small. I will be working on large faces. A4 size.

My mental Asaro head

 John Asaro is a genius. He created the Asaro head.

I believe that perfecting Asaro head in all angles will make me fit for portrait drawing. 

My mental picture of Asaro head (with difficult planes) is approximately 96 percent correct. My hand is unable to follow. 

Some theory

1. Forehead upper line to eyebrow line; eyebrow line to bottom of nose line; and bottom of nose line to bottom chin are equal. Courtesy Andrew Loomis.

2. Lower lip line divides the bottom of nose line to bottom chin distance equally. Courtesy Youtube, Andrew Loomis.

3. The width of the lip is equal to and aligns with the position of the eye ball. Courtesy YouTube, Andrew Loomis.

I am adding one statement. I do not know whether correct or not. 

4. The error begins with the position of the ear. And that is where you have to start. And the error propogates causing planes of face. 

Try with point no 4 and tell me if I am correct. Also add some other stuff like point no 4 and let me know.

Best wishes dear readers.

Another Expression

 I am presenting another expression. It has been a while since I posted a blog post. Here we go :-

I am currently working on faces. Wish I was a student of an art school in USA, I would have learnt many concepts in art theoretically and could have practiced practically also. I saw the work of William Oberhardt in Google images. The name of William Oberhardt was suggested by Andrew Loomis in his book which I possess. Precisely that is what I want to achieve. To draw like William Oberhardt. But he is so far away. He is like amazing !! I have mastered Asaro head in all angles to 80%. I follow my own 3.33333 . . . . logic in drawing the Asaro head. It isn't a perfect Asaro head but a 3 part planes division both one axis and the perpendicular axis. In Asaro head I have progressed. But the quality of an actual face drawn is not okay. The imagination, different face shapes, the lines do not happen. It is only 20%. Maybe some reader would give some inputs on improvement.(The desire is to be a Tom Richmond too. But the desire to be a William Oberhardt is very high).

Mathematics progress is stopped. There is not much to look forward too. Though I have put this equation above, the idea was with me since about two months. So mathematical expressions have died. I am thinking on the lines of linking Taylor series to Rational number series. These two, that is Taylor series and Rational number series, I want to link up-to Relativity. Many may think it is utter madness  and makes no sense. But that is how I think. When you have a field you have Taylor series. In the absence of field that is in space you are dependent only on your thoughts. The Rational number series. And when we talk of space we think about relativity.So that is what is the connection. Hope you have understood my view point. 

I hope I am not belittling the greats. The greats like William Oberhardt, Tom Richmond or Brook Taylor. It is just that I am paying them respect in my own way. I have great regard for them.

Hope India is able to contain Covid pandemic. The daily number of cases have reduced. But winter is about to arrive. I hope Covid is contained. Let us improve cleanliness and discipline. 

I wish a good day to all my blog readers.

Try these expressions

 

You may try these expressions. Or just observe them. And try to get the feel. These expressions are great for logarithmic functions.

You may recall the work of Colin Maclaurin who extended the Taylor series and gave the logarithmic functions.

COVID-19 pandemic is not dying. It is difficult in a country like India where there is not much cleanliness and discipline. A much much bigger role is expected from the societies and each one of us. I wish all of us take on a drive on war footing and defeat this pandemic. War footing means sacrificing everything we have just for this cause of defeating this COVID-19.

Have a nice day, dear blog readers.

Wishing to state

 

Final Formula

 I wish to present the following equation. It is a general formula and I call it two sets of consecutive numbers. If used with three fingers it is 1,2 and 2,3. If used with four fingers it is 1,2 and 3,4, If used to more than three or four then there will be a break. However, the formula holds.

The formula is

I do not know whether there is further scope to this concept.

Have a very nice day, dear blog readers.

(I shall be checking on this further. And in case of error I shall write about it. In case there is no error then this blog post shall remain unchanged.)

It is indeed nature !!

 I have been advocating "Rational Number Series" for two or more years. I claimed that n/(n+1) as nature.

I have a point. In fact 3 points. First is the finite series which gives 2n/(2n+1). Second is (n^k)/((n+1)^k) as infinite series. Third is ((-1)^k)(n^k)/((n+1)^k) as infinite series. The second and third points give easily expressible numbers. See the screenshots below.

I was right in advocating and popularising n/(n+1). 

Two presentations

 I am presenting two equations below.

Hope you like them.

The sway and the three joints four fingers theory

 My three fingers and four fingers theory and its extension has given many algebraic expressions to our group of institutions.

I have another theory. And that is the sway of legs and hands. And the three joints and four fingers thing.

I equated these two. Got y as 2/((2n+1)(2n-1))

The initial results are promising. I did infinite sigma on it. Got 1.

I did infinite sigma with alternating minuses. Got pi.

I did finite sums. Got a great and easily expressible answer.

Here are the screenshots.

A few more . . .

 I am able to make expressions with lot more ease now. Below I have pasted some expressions.

I would like my readers to try out all these expressions. You may have free time too. Because of COVID-19. 

Please also consider using 'Rational Number Series' in your thoughts. (n)/(n+1) is what I mean. You may refer to my expressions too. I believe it is nature.

Try this one too

 

I am quite convinced about this combination expression that I have put. Because it was in 2010 that I made the Measuring Behavious equation. I did not know many things. I was not sure of comb(n,r). But absolutely sure of comb(2n,n). And now comb(n,1). comb(n,1) was a genuine thought inside me. I used to think why only one aspect? 

So, I can use comb(n,1).