## Intro to comfort science a.k.a ergonomics

“The world can be a different place once you start observing and interpreting the most important facts”….(someone said it).Isn’t it amusing that almost everyone on a computer chair finds it tough to understand what that knob on the bottom does?Engineers have actually made a science out of comfort and muscle reactions and what-not and it plays such an important role in our everyday static lives.I spend almost 12 hours sitting in one place and after enough years of avoidable pain,my mind decided to write this blog.(Office chair Buying Guide – John Lewis (johnlewis.com)

Almost everyone must have read the 3 laws of Newton.

MUSCLES CAN PUSH BACK

I found this simple and amazing explanation of what happens to a runner at the blocks,

First,there’s an intro to internal and external forces.External force is when you shub the athlete fr

English: Diagram of contraction of smooth muscle fiber (Photo credit: Wikipedia)

om behind in a bid to win your bet.Internal force is generated by the contraction of the muscles of the athlete himself.

There’s a small explanation to why the Earth isn’t affected by all this dashing around,” Applying Newton’s 3 rd Law, The performer cannot move the earth but receives significant acceleration ,This is called Ground Reaction Force Action force of muscle contraction- Equal and opposite force”.

Here comes the main part:

“Using Newton’s Laws one can explain how an athlete accelerates out of the blocks at the start of a race. The athlete remains at constant velocity, at rest, in his blocks at the start of a race due to Newton’s First Law – the Law of Inertia. In order for him to accelerate an external force must be applied. As the athlete uses his muscles to generate a force into the blocks/ground there will be an equal an opposite reaction force pushing him forwards, due to Newton’s Third Law – the Action-Reaction Law. This resultant force is the external force required to overcome the inertia (Newton’s 1st Law) and the athlete accelerates from the blocks. The acceleration of the athlete is in direct proportion to the size of the resultant external force due to Newton’s Second Law – the Law of Acceleration. The acceleration can be calculated using the formula F=ma.”*1*

CONCLUSION TO THIS ENLIGHTENING ARTICLE?

So,the next time you design a chair,take into account muscle reaction,it takes way more effort to lift those hefty arms after 12 yours,especially when the arm rest is non-existent and takes a lot of effort to lift-up that sleepy head,weighs something like 15 pounds(with 1.7 kilos of jelly inside :p).

Either be smart or take-up a construction job.

SIDE NOTE:

Now,if you were paying attention,you must have asked the question,”Aren’t muscles supposed to expand to produce a force and hence a movement?”Wrong.

“Muscles work together by having one muscle contract while another extends or relaxes. The muscle that contracts is called the prime mover.  But in order for this to happen, there is another muscle involved. It is called the antagonist. This muscle has to relax to allow the movement. If the antagonist doesn’t relax, then the part that is being pulled would not move, but remain rigid.”*2*

References:

I found this amazing site on ‘MUSCLE CONTRACTION MECHANISM’.

http://meat.tamu.edu/muscontract.html

## Banks,accounts and interest rates

BANKS,ACCOUNTS AND INTEREST RATES

The people working at banks are stressed-out most of the time.,and if your’e working in a country with  1 billion people,this job is not easy.So,the next time you get into a bank with no idea about the type of account you want to open,or the best loan for you,don’t expect to get clear answers from desk-clerks.I’m gonna attempt to provide a ‘BRIEF INTRODUCTION'(this should be a marketing phrase,one always ends-up wanting to know more).I am no bank manager,i learnt all this from observation.

ACCOUNTS:

1.SAVING ACCOUNT:

This type of account is mostly for employees who ‘withdraw’ frequently.The interest rate is not going to be as high as a fixed-deposit account,the reason being banks can’t lend out too much of your’e money because of policies set by the regulatory body.(Wanna know more?check the References).The interest rates on Indian savings accounts are usually 3-5%,depending on the PLR(Prime Lending Rate) set by the RBI + the rate the bank sees fit.(When you see a clause in the newspaper which says,”RBI has de-regulated savings bank interest deposit rates”,this means the bank can set any rate they want,depending on their competition ofcourse).

So,the next time an Indian bank manager ask you,”Which type of account do you want to open?”,for Heaven’s sake,don’t reply,”I hope you could help me with this”.The worst case,he’ll chuck you out with a Current account :p).-the interest rates on current accounts are usually 0%.

2.FIXED DEPOSIT:

These are the highest paying,right now at 8.5-9.5%.Once you put your’e money in a fixed deposit account,its essentially in ‘Lock-up’.You can’t take it before it ‘matures’.If you do,there’s going to be a hefty penalty.Now,here’s the part your’e gonna get confused,read carefully:

The fixed deposit interest rates are lower for accounts with longer maturity periods,for ex.a F.D a/c with a 1 year maturity period will have a interest payback of 8.5% say, and a F.D a/c with a maturity period of 2 years may have a interest payback of 8%,let’s say.Now,intuitively,you must have thought the bank was supposed to pay you more when you keep the money ‘IN’ for longer right?Nope.

Reason:Essentially,you want the bank to be paying you an 8% on your principal constantly for 2 years straight,no matter what the market situation is.Perhaps,in the coming next month,the interest rates would be dropped to 7%,but your’e still gonna get the ‘good end of the stick’ i.e 8% because you were clever enough to open an F.D at the right time!

3.CURRENT ACCOUNTS:

This type is mostly for business-men/women.Banks rarely pay any interest on it.But,the pros of such account are that:

1.The bank certifies that you’ve got enough balance in your account(something that’s gonna give your customer confidence).This is kind-of like using the ‘Cashier’s check’.

2.You get to use the overdraft policy i.e draw more than you have in your a/c but upto certain limits of-course.

Remember,banks can charge any interest rate they want depending on your credit points.Investopedia explains this well,”Default risk is the main determiner of the interest rate a bank will charge a borrower. Because a bank’s best customers have little chance of defaulting, the bank can charge them a rate that is lower than the rate that would be charged to a customer who has a higher likelihood of defaulting on a loan

<U>References:</U>

1.If you want to delve deeper into why the interest rates are so low: http://www.nytimes.com/2010/12/04/your-money/brokerage-and-bank-accounts/04money.html?pagewanted=all

2.How Do Banks Change Rates Based on the Prime Rate?

## Growing exponentially

Leonhard Euler’s signature (Swiss mathematician),the first occurrence of the dreaded ‘e'(Photo credit: Wikipedia)

You often hear/read this phrase “your savings will grow exponentially”.But,there is a formula to calculate the capital resulting from the principal P invested for 1 year.This is given by:

$CAPITAL=PRINCIPAL*(\lim_{n \to \infty }(1+\frac{r}{n})^n)$,

where r is the annual interest rate and n is the equal intervals a year can be divided into.Obviously,when n tends to infinity,we are talking of a very very small time interval.

Amazingly,the above formula becomes

$CAPITAL=PRINCIPAL*exp(r)$ and this is independent of the infinite equal intervals of the year n.

So,Euler found out that

$exp(r)=\lim_{n \to \infty}(1+\frac{r}{n})^n$,NOTE that this formula is valid even for complex numbers.

Its easy to verify if you expand using the binomial theorem,but put yourself in Euler’s chair and you’ll soon find that this would require a rigorous mathematical proof,something that most mortals dread.

The exp(r) is called the exponential function.I’ve overheard friends mistakenly call this the Euler function.In-fact,there exists a Euler function but I don’t understand the math behind it.So read-up and help me!(http://en.wikipedia.org/wiki/Euler_function)

There also exists a Euler phi function.Euler came up with this new function when he was trying to generalise Fermat’s little theorem(if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p).You can think of   $\large \phi(n)$ as being the number of +ve integers less than n and relatively prime to n.This function makes understanding the little theorem more easy.

The exponential function also has its inverse function-natural log function.The beauty of the inverse function is that undoes the action of the original function,reverses direction might be more apt.The log function interestingly has a +ve real number domain i.e it doesn’t exist for -ve real domains.The reason for this is because of the base we use for the natural log.It can’t be imaginary.The number x to which we raise the power r to,cannot be negative when we define the exponential function.Simple!

## Obsessive art-bursts of creativity

I started-off on landscape sketching 2 weeks back. An American painter in the nearby village helped me with some art classes,”drawing experiments”, she calls it. I was so excited that I mistakenly picked-up a the wrong pencil to sketch.After an hour of sketching the pond with moving cattle nearby,I rushed back to her room.She replied,”I can hardly see the strokes,you picked the wrong pencil!”she chuckled. Aah!I looked at my watch and surprisingly I had been out for more than an hour. I was so engrossed into drawing that I forgot time.

Anyway,she suggested that I pick-up the charcoal and start outlining the boundaries.After I was finished,it looked so beautiful( to my eyes :p).What I attempted to do was try and put ‘projective geometry’ into practise.

I couldn’t stop drawing that whole day.I drew the rocks on the beach,the buildings nearby,my dog,I just loved it.I even dreamt of drawings,I woke-up at 2A.M  and drew what I dreamt and the most astonishing thing happened.When I tried to recollect what the dream was,it was all fuzzy.But,when I started drawing,it came out just like it was.When people dream,they’re always the observer(3rd person),in the sense,they can themselves from the outside,like a movie .Just to be sure I was right,I grabbed Freud’s “Interpretation of dreams” and checked.I wonder what would happen if I realized everything was a dream when I dreamt.

The obsessiveness stopped in 24hours.If I was in a Western city,some Psychiatrist would claim this as abnormal and prescribe a pill,which I think is ridiculous for such short bursts.Sketching is a motor activity,so it must have something to do with the basal ganglia but you can control it or let go if you want.You don’t need a pill for everything!!

Every person in the world has these bursts of creativity.For some people,it lasts long enough to get recognised,some can’t control this and society pushes them aside.I’ve had such bursts with math problems too.The other day,I was trying to understand the line integral(math jargon),I dreamt of the familiar wine-glass tone producing experiment and tried to figure out the work-done using the line integral.I might have got the results wrong but the puzzle was already firmly planted in my mind.So the next time I sit idly in a cafe’ and have nothing else to do,I’ll solve the puzzle or maybe draw something in front of me.

Have fun sketching,painting,whatever.Don’t follow the crowd,its not logically correct you see.

## Fractals and coastines

An example of the coastline paradox. The coastline of Britain is measured using Fractal Unit = 200 km, then the length of the coastline = 2400 km (approx.) (Photo credit: Wikipedia)

Unit = 50 km, length = 3400 km (1000 km more) (Photo credit: Wikipedia)

How can one approximate the length of a country’s coastline?It’s not intuitive because of the way we choose to measure curves.

Curves can be approximated by straight line segments.This is easy to carry out for a curve which follows a regular smooth shape,but the curves on the coastline have varying curvatures everywhere(‘JAGGED GEOMETRY’).

“So,what’s the problem with using line-approximation here?”

Take a look at the relation between the number of line segments needed to approximate a curve -‘N’ and the length of the line segments ‘L’ inserted    :

$N\alpha \frac{1}{L}$

i.e To truly approximate the curve,the length of the line segments have to be made super tiny and the number of line segments needs to be super-huge.This can be represented by

$\lim_{N \to \infty L \to }N*L=REALcurvelength$

As I was saying,this is all cool and dandy for a curve with a constant curvature.Now,take the following curve for example:

It seems logical to first line-approximate the more curvy curve C2 first and then use the same line segment lengths for the less curvy curve C1. Next,imagine doing this for a coastline. We end-up using these smaller line segments for approximating even the bigger curves(like C1) and this would imply the product N*L diverges.Why does this product diverge?

Because,even after L has reached its optimal minimum,N has to increase even more.

So,in-order to take care of this divergence,mathematician by name Mandelbrot came-up with a different proportionality relation,(which took care of the different wiggles like the coastline problem).

$N\alpha \frac{1}{L^D}$

How D can be ascertained is very nice. A scientist by name of Lewis Fry Richardson plotted the graph between the ‘length of a country’s coastline’ and ‘the scale size’ used to measure this length. This is what I tried to explain above,the country’s coastline seems the same length no-matter how much you zoom-in or zoom-out provided you use the same scale size!This seems very non-intuitive to me.

Conclusion: So now,we can define a coastline just by one parameter-D.For example,D for the coastline of South Africa is 1.24.

http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

Book:Chaos,Fractals and self-organisation by Arvind Kumar

Illustration of offset yield point. Key: 1: True elastic limit 2: Proportionality limit 3: Elastic limit 4: Offset yield strength, usually defined at e=0.2% S: Engineering stress e: Engineering strain A: Undeformed cross-sectional area F: Uniaxial load L: Underformed length l: Elongation Reference: G. Dieter, Mechanical Metallurgy, McGraw-Hill, 1986 (Photo credit: Wikipedia)

So,I’m reading-up on engineering metallurgy stuff and there’s something called the ‘stress-strain’ diagram for different kinds of materials.Now,I kind-of had a connection established between fractals and one of the plasticity measuring theories.

Have a look at the diagram on the left.2 is the point where the linearity ends.3 is the point where the first occurance of plasticity occurs.Now,the first occurance of plasticity is measured by a sensitive instrument.So,for a super-sensitive instrument,the first occurance of plasticity appears before-hand.So,the point 3 is essentially determined by the quality of the instrument,just like scales determine the length of a coastline.

But,I’m not having time to go further into the theory.Maybe,some mathematician/engineer out there can think this through.

## POLYNOMIALS AND CHESS

Everytime I fall into trouble in explaining math,I remember this quote,”If you can’t explain it simply, you don’t understand it well enough.” by Einstein and then run-back to the desk.

I have decided to celebrate the onset of the monsoon with a post on ‘rook polynomials’.Rook polynomials give you an idea of the number of ways to place 1 or 2 or k rooks on a mΧn dimensional chess board such that no 2 rooks are in the same row or column.What makes this combinatorial problem easier is that rooks move in straight lines.But first,I’ll have a go at an explanation of what are polynomials.

Rook-Moves (Photo credit: Kevin Byrom)

## Polynomials:

A polynomial is something that can be written like this Since polynomials follow a definite pattern,mathematicians have come-up with various ways of classifying them.One such is using the degree.The above polynomial has ‘degree k’.

I find one type of polynomial very interesting,this is the ZERO POLYNOMIAL. It can be written like this  where P(x) has all its coefficients equal to zeroi.e this is literally the number ‘0’.

Now,why would mathematicians go through all the pain of classifying the number 0 as a polynomial?

This is because the number 0 can be thought-of as the additive identity in the additive group of polynomials i.e when I add 0 to any polynomial,I will get the polynomial itself.This brings us to another question:If the number 0 is a polynomial,what should its degree be?

The zero polynomial is the ‘identity’.

But,take for example the math function.This is not a polynomial!Why is this not a polynomial? Because,log(t) is what is called a ‘transcendental function’.Transcendental functions ‘transcend‘ algebra in the sense they cannot be expressed by a finite number of terms. Functions are split into 2 types:transcendental and algebraic: I’ve written more on these functions at the bottom of the page. So,to recap,transcendental functions cannot be expressed as a polynomial because they have an infinite number-of terms.

Polynomials can also approximate functions,this is one super-awesome property engineers use to great advantage.These polynomials are called Taylor polynomials.Don’t confuse them with Taylor series!!Taylor series extend all the way to infinity if the function lets you do so.

Oh yeah!If you’ve lost your calculator and need to calculate the square root of a number,guess what your’e going to using then??

## Rooks:

Most people know of ‘rooks’ in chess,they can only move in a straight line,no zig-zags are allowed. Lets create a math puzzle where you have to find-out the number of ways in which you can place k rooks on the m*n chess board,so that,no 2 rooks are on the same row or column, in other words,no 2 rooks can attack each other.This is a combinatorial problem where we could use variable ‘x’ to help us denote the number of rooks.

To help with the problem,Barbaeu suggests a delivery truck problem.If you can’t read the name of the cities,just look at the first letters.This can be viewed as a everyday logistics problem involving transportation of goods from one city to another but with constraints like:

1.restrictions for certain city-city transits(marked by X’s)

2.only 1 truck can be sent each source city.

In the below ex.,trucks can’t travel from Mumbai to Kolkata and so on.Your task is to find all possible ways of transporting with 1 or 2 or 3 trucks between cities in any given day.Note that the maximum number of trucks you can assign for 1 day here is 3 which can be found as the min(3×4).

Now,use the same principle for your chess board problem,except now the X’s are the places where the rooks are placed.

Note that by ‘placing rooks’,i mean ‘placing non-threatening rooks’.

Now,then,in how many ways can you place 0 rooks on the board?Think about it carefully,its 1.

In how many ways can you place 1 rook on the board?Obviously 16.

In how many ways can you place 2 rooks on the board?

1.Choose 2 particular rows.This can be done in 4C2 ways.

2.Next,choose 2 columns in these rows.You will place the rooks at the intersection of row-column,but as given,make sure they don’t lie in the same column.This can be done in 2! ways.

Similarly for column:

1.Choose 2 particular columns.This can be done in 4C2 ways.

2.Next,choose 2 rows in these columns chosen.You will place the rooks at the intersection of row-column,but as given,make sure they don’t lie in the same row.This can be done in 2! ways.

Therefore,the number of ways to place 2 rooks in a 4×4 chess board is=4C2 x 4C2 x 2!

Append a x^2 with this value to know this numerical vaule was for 2 rooks.

Similar procedure for 3 and 4 rooks and use the same general formula like above.

These polynomials where the coefficients give the value of the combinations and the variables give info about the number of rooks is a type of 'generating polynomial'.

## More on algebraic functions:

Algebraic functions are kinda complicated because they take-up the form of a polynomial and their coefficients are polynomials in themselves.They(algebraic functions) can be represented by such relations:for which I can write the solution as . and this is what we finally get as a polynomial after rearranging and cross-multiplying:where,is itself a polynomial. For example, lets take this meaningless-looking complicated algebraic function

After rearranging,we get a nice-looking polynomial like this Transcendental functions cannot be split-up like above and hence do not arrive at a polynomial-like expression because there are infinite terms. Never ending stuff…..you cannot make a polynomial with a finite leading coefficient out of this. There are many ways of telling whether a function is algebraic or transcendental.This is when math starts to get fun!!

Answer for:What should the degree of a zero polynomial be?(Thanks to the versatility of physicsforums.com,I came to know of this topic) Say,we have 2 polynomials P and Q. We know that, Then,if polynomial Q were equal to the number 0, The only way for this to be possible is if the degree of the 0 polynomial were arbirtrary and negative-

## MORE FUN FACTS ABOUT POLYNOMIALS:

You can predict the end behaviour of polynomial curves just by looking at the power and sign of the leading coefficients.

For example,a polynomial  will have both its ends going upward like in the first diagram POLY1.

A polynomial like so  will have both its ends going downward like POLY 2.

A poly with the highest degree being odd will look like the 2nd figures,depending on the sign of-course.

** There’s something called the Horner’s method(o with German umlaut) which you can use to substitute the value of the variable x into the polynomial.You can actually do this faster than a calculator.

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