A probability question on Sudoku Grid

 What is the probability that given n trials to fill the numbers in Sudoku grid how many(X) valid numbers will filled at the end ?

What is the probability distribution for the random variable X ?

Currently working on ...

 Working with two scholars on two themes.

Theme1: Constrained PDE optimization

Theme2 : Cortical surface reconstruction using level sets.

Theorems on Roots of Polynomials in Geometry, Topology, Complex Analysis and Functional Analysis

 Roots of Polynomials
Prof. V. Kannan
Dept. of Mathematics
SRM University-AP
23 February 2022

Abstract: "We discuss four classical theorems on roots of polynomials. They relate to four
different branches of Mathematics, namely, Geometry, Topology, Complex Analysis and
Functional Analysis. The theorems are stated below:
Theorem 1 (Lucas). Every root of the derivative of a polynomial lies within the convex
polygon formed by the roots of that polynomial.
Theorem 2: The roots of a polynomial vary continuously with its coefficients.
Theorem 3 (Cauchy): If all the coefficients of a polynomial lie in the unit disc D, then all
the roots lie in the disc 2D.
Theorem 4 (Bernstein): The norm of the linear operator D (namely the differentiation) on
(The space of polynomials of degree at most n on D) is exactly n."

Deterministic approach Vs Probabilistic approach

The time taken will be deterministic if you proceed by steps else it will be random variable.

Compute all the possible paths to reach from A to B in North or East or NE direction only.

 

Stone weight in your ornament formula - మీ నగల్లో ఉన్న రాయి బరువెంత ?

d1 and d2 are densities of gold and the stone, resp.

densities can be found by the standard density tables online for various stones and gold types.

M and V are the total mass and volume of the ornament.

Mass is easy. 

Volume can be easily found by dipping in water in a graduated cylinder or volumetric glass of chemistry lab.
 

Non linear mapping of rectangular grid to sinusoidal grid.

 

Why Mathematics is the Beautiful language of all and will be.

1. It has a precise meaning for every expression it defines, where as the other languages do not.

2. You can add different expressions which you can not do with other languages. Ex: adding laughter to depression.

3. It is well understood by any computer in the world.

4. A french mathematician can speak to a Japanese mathematician without using french or japanese and most importantly without any misinterpretations. 

5. It uses the least number of letters or symbols to express any complicated situation/system.

6. There are no bad words in Mathematics.

7. There are no censorship in mathematical language.

8. Nature better expressed to finer details better in Mathematics than any other language,. 

9. It is open to talk to any other branch of science.

10. It can still provide a sustained employment for decades to come.

11. No sentence in Mathematical language is outdated at least from 500BC(Euclid's poems on Geometry)

Converging Geodesic Centers for Andhra Map

The convergence happened only if time step in the evolution is at most 1.

If more than 1 then the scheme is blowing up.

Moment of triumph in the time of Covid-19 research Honeymoon

Did not expected that the glitch in the procedure is a matter of "same" or "full" in the filter2 function and with bwdist function.
Eureka !

Daily Covid-19 cases rising no shorter than an exponential with parameter of 0.04

Doubling rate of COVID-19 in India

5-day moving average spread of COVID-2019 in India

Doubling Rate calculations of Covid Patients

If you are looking for daily parameters on Doubling rate go here 
The following is as per the govt officials declaration and mine is here under in a graph with 11.39 days against 11.3 of Union Health ministry, as on 29th April 2020

దేశంలో కరోనావైరస్ కేసుల రెట్టింపు రేటు ఏప్రిల్ 29వ తేదీ నాటికి 11.3 రోజులకు తగ్గిందని కేంద్ర ఆరోగ్య శాఖ మంత్రి హర్షవర్థన్ తెలిపారు. ఏప్రిల్ 20వ తేదీ నాటికి ఈ రేటు 7.5 రోజులు అని, మార్చి 25వ తేదీ.. అంటే దేశంలో లాక్‌డౌన్ అమలు చేయకముందు 3.4 రోజులు అని కేంద్ర ఆరోగ్య శాఖ తెలిపింది.

కోవిడ్-19 సోకినట్లు నిర్థరణ అయిన కేసుల సంఖ్య ఎన్ని రోజుల్లో రెట్టింపు అవుతోందో ఆ రోజుల్ని బట్టి ఈ రేటును లెక్కిస్తున్నారు. ఏప్రిల్ 20వ తేదీ నాటికి ఈ రేటు ఆంధ్రప్రదేశ్‌లో 10.6 రోజులు, తెలంగాణలో 9.4 రోజులు అని ప్రభుత్వ గణాంకాలు చెబుతున్నాయి.

భారతదేశంలో తొలి కోవిడ్-19 కేసు 2020 జనవరి 30వ తేదీన నమోదైంది. అప్పట్నుంచి 44 రోజులకు వంద కేసులు నమోదు కాగా.. తర్వాతి వంద కేసులు ఆరు రోజుల్లో, ఆ తర్వాతి 200 కేసులు కేవలం రెండు రోజుల్లోనే నమోదయ్యాయి. అనంతరం ప్రతి నాలుగైదు రోజులకు ఒకసారి కేసుల సంఖ్య రెట్టింపు అవుతూ వచ్చింది.

ఏప్రిల్ 9వ తేదీ నాటికి దేశంలో కేసుల సంఖ్య 6400 దాటగా.. అవి రెట్టింపై 12800 దాటడానికి వారం రోజులు పట్టింది. ఆ కేసులు రెట్టింపై 24600 కేసుల్ని దాటడానికి ఎనిమిది రోజులు పట్టింది. ఏప్రిల్ 30వ తేదీ ఉదయం 10 గంటకు మొత్తం కేసుల సంఖ్య 33,050కు చేరింది. ఐదు రోజుల్లో కేసుల సంఖ్య 35 నుంచి 40 శాతం పెరిగింది. అంటే, రెట్టింపయ్యేందుకు అంతకుముందుతో పోలిస్తే ఎక్కువ సమయం పడుతోంది. ఈ సమయం పెరిగిన కొద్దీ రెట్టింపు రేటు తగ్గినట్లు.

Relation between Stock price and option prices I understood now !

After an amount of time and experiments with money only I understood the relation between stock price and its derivative value . Also I got an understanding over the time decay and volatility effect on the option price.
 With this I suggest everyone always short options when nearing the expiry .
And the shorting is very effective than buying.

Ex: If you the stock is growing then short the put.
Ex: If the the stock is losing then short the call.

The point here is that short on the out of the money option with a good theta.
The dangerous factor lurking here is the  beta, i.e. volatility.

Types of Calculus

1. Leibniz Calculus, derivatives-functions
2. Newton's Calculus , Fluxions-Fluents
3. Lambda Calculus, Computer science
4. Ito Calculus, Random Processes-Finance models
5. Variational Calculus, first variation-functionals
6. Propositional Calculus, Logic

Two volunteer problem in a circular campus

Octave/Matlab Code for the solution :

Blue and Red pair stands for the positions of the Two volunteers in a given circular campus.

For various  random pair of seeds the optimization finally resulted in a set of four nonlinear integral equations.
 Solved them using iterative procedure.
The locus of the volunteer positions have come to be on a particular radius(=0.42676) circle with each one on antipodal positions.
  Later, based on the antipodal property observed from the computations the value of the radius was found to be exactly 4/(3pi)=0.42441 by evaluating the integral equations in polar form for the unknown radius.


Code starts here :


%%% Two volunteer selection problem

T=[0:0.01:2*pi];
R=[0:0.01:1];
[GridR, GridT]=meshgrid(R, T);
iX=GridR.*cos(GridT);
iY=GridR.*sin(GridT);
dA=GridR*0.01*0.01;

for ini=1:100
%%initialization
P=[rand*cos(rand*2*pi), rand*sin(rand*2*pi)];
Q=[rand*cos(rand*2*pi), rand*sin(rand*2*pi)];

 for it=1:40
   dD=sum(P.*P-Q.*Q ) ;
   sgn=sign(2*(Q(1)-P(1))*iX+2*(Q(2)-P(2))*iY+dD);
   %% x0, y0
   isgn=1-sgn;
   Nume=sum((iX.*isgn.*dA)(:));
   Deno=sum((isgn.*dA)(:));
   P(1)=Nume/Deno;
   Nume=sum((iY.*isgn.*dA)(:));
   Deno=sum((isgn.*dA)(:));
   P(2)=Nume/Deno;
 
   %% x1, y1
   isgn=1+sgn;
   Nume=sum((iX.*isgn.*dA)(:));
   Deno=sum((isgn.*dA)(:));
   Q(1)=Nume/Deno;
   Nume=sum((iY.*isgn.*dA)(:));
   Deno=sum((isgn.*dA)(:));
   Q(2)=Nume/Deno;
 
  endfor
   polar( atan2(P(2), P(1)), norm(P),'r*');
   hold on;
   polar( atan2(Q(2), Q(1)), norm(Q), 'b*');
   pause(1)
   endfor

Trial with Google form based quizzes

Preparation of the form is taking time.
First I typed all the questions on latex and produced pdf.
Then used snipping tool to cut the questions into images.
I just used a b c d options for all the questions.
Some of them are of one answer and some are with multiple answers.

I shortened the quiz URL using bit.ly.
And, I made blocks of 3 questions as a section.
Prepared 7 sections out of which I asked to do 4 sections in a class and the rest are for next day.

After the quiz is finished I looked at summary of the responses and released the scores and answers only after everyone is done with.

In my attempt with a section, I accidentally ticked a setting to view the missed questions which released the remaining all questions after their scores released in one of the classes.

I tried to reveal the answers after the quiz but they were not appearing as I thought.
I am missing something in this.

There is no auto save for the forms when the quiz is going on. This made

భీమ్ bhim upi password setup

Chose fibnonacci series starting with any two integers.
If you reverse the order of initial two digits then the sequence will change completely.
Ex. 011235
       101123
        123581
          213471
           134711
             314591

Books that helped me to reach IIT kanpur

1.Thomas and Finney, Calculus and Analytical Geometry.
2. Angus E. Taylor Advanced Calculus
3. History of Mathematics, Carl B Boyer and Uta C Merzbach
4. Integral Calculus, Maity and Ghosh
5. Vector Calculus by Shanti Narayan
6. Inequalities by P. Korovkin Mir publishers
7. NCERT books Differential equations
8. Unified publishers books for BSc.
9. Mathematics today weekly magazines
A. Central University of Hyderabad entrance previous papers.
B. Schaum's series book on basic algebra.
C.