Mathematician's Apology

Hardy's Mathematician's apology figures among the must read for any Math major. The book was written when Hardy was over 60 and passed his prime. The book is a fast read and is divided into 29 sections . It begins with a lengthy forward by Snow detailing his accquaintance with Hardy and tidbits about the book. From the beginning Hardy is candid that it is his personal opinion and inspite of inputs from his reviewers remains firm in the stance he has chosen. Numerous books have been written since Hardy about what constitutes a beautiful mathematics. Hardy bites more than he can chew when he says applied mathematics is elementary and real mathematics is not. But then it was before the development of Nuclear bomb and advent of computer sciences. Hardy acknowledges that its not easy to keep defending his assertion he made in 1920s about real and applied mathematics.
He excoriates both Alfred Northwood and Hogben the other defenders of mathematics. He is especially critical of Hogben and I am sure most Mathematicians will readily identify with Hardy here.
It is hard to defend ones choice without defending oneself. Hardy comes out both proud and humble in this book. Its not a motivational book that would propel you into plunging heads down into mathematical water. He reminices about his collaboration with Littlewood and Ramanujam. His continued lamentation about getting old is unsettling when people like Avraham Trahtman continue to do ground breaking mathematics in their 60s. Hardy's disdain about expository mathematics appears sophomoric especially if you have read Paul Halmos "I want to be a Mathematician". Its one thing to come up with something and its other to make it accessible and provide insight into it. A great mathematical exposition kindles the interest of many otherwise many results gather dust.

Grading and other stuff

I am updating this from Morris Library. I came here to do some transaction. I am glad that today I was able to return the corrected midterms for my other two classes. I felt like I was procrastinating this for a while. I realize that I need to do more math for fun and learning and somehow I am getting caught up with things which aren't so important. So here I am at library I will check some journals before going home

One Minute

I read one minute manager few years back and yesterday before going to bed I pick that book again and I was pleasantly surprised that there were things in the book which I was using in my life while interacting with other people. One minute manager is one of the quickest read. The story has 3 main ideas
1. One minute goal
2. One minute praise
3. One minute reprimand
The idea behind one minute goal is to write your goal on a single sheet of paper. One minute praise is to catch people doing things right in the beginning and one minute reprimand comes later when people have been coached and have become independent and then they do mistake. When reprimanding such individuals one should be careful to point out their mistake and not making any judgement about the person and then letting the person know they can do it and leaving them on positive note.

Good Morning

Its 10:11 and I am in the JALC library updating this blog. I had a morning class today and surprisingly Mark was there to do the evaluation. I had a review sheet ready for the exam and then I covered the last topic. I have many good students and they are steadily improving. I am planing to give a speech today. It has to be informative and I still haven't decide what should be the exact topic. I considered talking about "Erdos" or talking about "How to solve it" or "Ramsey theory" or "Pigeon hole principle". Each of these are interesting topics and can elicit an aha response. Here is something I found that can be used as an online octave.

Calming

There is nothing more calming that working out few problems. I was looking at this problem that appeared in the recent Pi-Mu journal. Find the int(ln(1+x)/x,x,0,1). One easy way to solve this is using the series of ln(1+x)/x and then you end up with an alternating series which closely resembles bessel problem and then its easy to see that the even terms sum to bessel function. Check the solution here

Rambling Pascals' Wager

I love teaching and one of the reason is we all like to flaunt our opinions, however if you are careful you can see that there are many times you end up saying things which you have cursorily thought about and then your student comes up and say that it was a good idea and then you go back to see why was it so interesting. I remember mentioning Pascal in my class while introducing Pascal's triangle. A brief comment on his work on making a mechanical calculator. His contribution to math and religion. I even mentioned Pascal's wager and why it doesn't even amount to anything and then one of my student tells me that he did checked out Pascal's wager and found it fascinating. I read pascal's wager in Poulos book called irreligion and its one argument that many people would like to foist on you. They will tell you that how benevolent their god is and then how much he likes to punish people who won't believe in him. Now if you observe many of these people they don't really believe in what they say because their actions most of the time belie their belief and on one side when they say omnipotent god and on other side they have no compunction in fooling their on creator. Anyway I think I am digressing.
The pascal wager is "If god exists and you believe in god then you will go to heaven if god doesn't exist and you believe in god you have nothing to loose but if you don't believe in god and he exists then you are going to hell and you have everything to loose". I did a quick search on internet and ended up with this argument which has been my guiding philosophy when I made peace with the god.
"It is better to live your life as if there are no Gods, and try to make the world a better place for your being in it. If there is no God, you have lost nothing and will be remembered fondly by those you left behind. If there is a benevolent God, He will judge you on your merits and not just on whether or not you believed in Him."

Fractal solution, Plotting complex plane in Matlab

The following is a solution set for a function f(z) =z^4+8iz^2-25. The roots of the function are
2-i, -2+i, -1+2i and 1-2i
The color shows which values will converge to the solution in the complex place


The complete matlab program is as below


% Using Newton Basin method to plot in complex plane
% The complex function is z^4+8iz^2-25
% There are 4 roots 2-i, -2+i, -1+2i and 1-2i of the above function
% It shows how starting with different points root converge

tol =.01;
a=0;
b=2;
c=0;
d=2;
m=1000;
n=1000;
x=linspace(a,b,m); % Setting up the x axis Coordinate System a to b
y=linspace(c,d,n); % Setting up the y axis Coordinate System c to d
z1=2-i; % root 1
z2=-2+i; % root 2
z3=-1+2i; % root 3
z4=1-2i; % root 4
lmax=20;
r=ones(n,m); % Setting a n*m matrix to be all 1
for j=1:n
for k=1:m
z=x(k)+i*y(n-j+1); % Generating the complex number, Note n-j+1
if z == 0;
z = tol;
endif

for l = 1:lmax % Newton iteration
zz = (3*z^4+8*i*z^2+25)/(4*z^3+16*i*z); % Newton's Formula for Iteration
if abs(z-zz) < tol
if abs(z-z1) < tol
r(j,k) =1; %Red
elseif abs(z-z2) r(j,k) =55; %Pink
elseif abs(z-z3) r(j,k) =33; %Light Blue
elseif abs(z-z4) < tol
r(j,k) = 42; %Deep Blue
endif
break
else
z=zz;
endif

endfor
endfor
endfor

colormap('hsv')
image(r)
axis square
axis off

Happenings

I am posting this from Carbondale Library. I am just coming after attending the picnic for GLBT held at Church of Good Shepherd. It was great to see Hugh and Judy there and I think overall the picnic was great. They had plenty of food and I saw a lot of familiar faces in Carbondale. I also had time to do an hr long sitting with Jamie and Katherine today. We decided last week at Co-op to do it and everyone was glad we did it today. Yesterday I pretty much finished the Math book. Its more like a coffee table book with lots of interesting picture accompanying each article. It also had a butterfly curve. Which reads as exp(cos(x))-2*cos(4*x)+sin(x)^2/2. In the morning today I plotted the Newton Basin curve and that pic is awesome, I will be posting it here. The neat thing was it gave me an idea of how to plot complex numbers. The method is about finding which points converge and which won't converge to the roots of the polynomial with complex coefficients.