I was very pleased on reading this news that Government of India has decided to celebrate the upcoming year 2012 as the National Mathematical Year. This is 125th birth anniversary of math-wizard Srinivasa Ramanujan (1887-1920). He is one of the greatest mathematicians India ever produced. Well this is ‘not’ the main reason for appointing 2012 as National Mathematical Year as it is only a tribute to him. Main reason is the emptiness of mathematical awareness in Indian Students. First of all there are only a few graduating with Mathematics and second, many not choosing mathematics as a primary subject at primary levels. As mathematics is not a very earning stream, most students want to go for professional courses such as Engineering, Medicine, Business and Management. Remaining graduates who enjoy science, skip through either physical or chemical sciences. Engineering craze has developed the field of Computer Science but not so much in theoretical Computer Science, which is one of the most recommended branches in mathematics. Statistics and Combinatorics are almost ‘died’ in many of Indian Universities and Colleges. No one wants to deal with those brain cracking math-problems: neither students nor professors. Institutes where mathematics is being taught are struggling with the lack of talented lecturers. Talented mathematicians don’t want to teach here since they aren’t getting much money and ordinary lecturers can’t do much more. India is almost ‘zero’ in Mathematics and some people including critics still roar that we discovered ‘zero’, ‘pi’ and we had Ramanujan.
Mathematics is beautiful and there is no place of ugly mathematics in this world. Mathematics is originated from creativity and it develops with research papers. Research Papers aren’t only very detailed and tough to understand for general student, but also interesting. Here, I have collected the list of some excellent articles and research papers (belong mainly to Math) which I have read and are easily available online. The main source of this list is ArXiv.org and you may find several research papers on ArXiv by visiting http://arxiv.org/.
If you know any other paper/article which you find extremely interesting and that is not listed here, then please do comment mentioning the article name and URL. Papers/articles are cited as paper-title first, then http url and at last author-name.
[It is better adviced to open these links to a new tab/window for smooth reading. ]
Cambridge Digital Library had made Newton’s exceptionally great works online.
Some times ago they added the Trinity College Notebook by Isaac Newton, which he used to teach in the college in 17th century.
List of other works of Newton can be found at www.newton.ac.uk/newton.html.
Dr. SMRH Moosavi has claimed that he had derived a general formula for finding the -th prime number. More details can be found here at PrimeNumbersFormula.com and a brief discussion here at Math.SE titled “Formula for the nth prime number: discovered?”
SOME MORE EXCERPTS ARE HERE:
This is the last month of the glorious prime year 2011. We are all set to welcome upcoming 2012, which is not a prime but a leap year. Calendars have very decent stories and since this blog is based on mathematical approach, let we talk about the mathematical aspects of calendars.
The international calendar we use is called Gregorian Calendar, said to be created by Pope Gregory XIII. Gregorian calendar was introduced in 80s of 16th century, to be accurate in ,—as a correction to earlier Julian Calendar. Julian Calendar was introduced by Julius Caesar and was based on the fact that there were days in a year, with leap year every fourth year. Astronomical calculations told us that one year on earth (the time required for the earth to complete an orbit around the sun) was equal to days —thus we can say that Julian Calendar hadn’t enough precise measure of dates. A difference of days per year meant that the Julian Calendar receded a day from its astronomical data every years (viz. Approx. years). More information on Julian Calendar can be found at http://en.wikipedia.org/wiki/Julian_Calendar .
The centuries old calendar came to an end as the accumulating inaccuracy caused the vernal equinox (the first day of Spring) to fall on March 11 instead of its proper day, March 21. The inaccuracy naturally persisted throughout the year, but at this season it meant that the Easter festival was celebrated at the wrong astronomical time. Pope Gregory XIII rectified the discrepancy in a new calendar, imposed on the predominantly Catholic Countries of Europe. He decreed that 10 years (11 March to 21 March) were to be omitted from the year , by having October 15 of that year immediately follow October 4. At the same time, C. Clavius proposed the scheme for leap years —which must be divisible by 4, except for those marking centuries. Century years would be leap years only if they were disible by 400. This implies that the century years are leap years, but are not.
A more detailed info about Gregorian Calendar is here: http://en.wikipedia.org/wiki/Gregorian_Calendar
There are many tricks to determine the day of a week for a given day after the year 1600 in the Gregorian Calendar. But we shall use a number-theoretic method to determine it, as described in the book ‘ELEMENTARY NUMBER THEORY’ by David M. Burton.
We all know that the extra day of a leap year is added to February month of the year, so let us adopt the convenient fiction that each year ends at the end of February. The months for any year Y are:
It is clear that if we count for any year , January and February must be in next year, of Gregorian Calendar.
We need another convenient notation as we denote days by numbers as:
The number of days in a common year is , and the number weeks thus are =52 weeks and 1 day while that in a leap year is claiming the number of weeks being 52 with two extra days. We could write last sentence as this way too:
Number of days in a common year is and that in a leap year is . [See FootNotes]
One extra day remaining after 52 weeks in a common year implies that the day proceeds for ‘one’ week-day for every year. February 28 is last (365th) day of a common year — it always falls on the same weekday as the previous year’s March 1 . But if it follows a leap year day, the last day February 29, its weekday is increased by two.
We can have a mathematical theorem to find which weekday a fixed date will fall.
THEOREM: The day with month , day , year , where (century) is equal to or greater than 16 and is any number between 0 and 99 inclusive, has a weekday number ,
where is the number chosen for corresponding month from the [LIST A], is the number which represent the date in common sense and the square bracket function represent the greatest integer less than or equal to .
After finding the numerical value of , we match it with [LIST B] .
Let me illustrate this with an example.
What day of week will be on December 9, 2011?
(That (today?) will be Friday off-course, but we are going to find it mathematically.)
For December 9, 2011:
(see list A)
This implies that the value of w to be but for being less than 7, we have . Comparing with table we get that December 9, 2011 occur on Friday (5).
1.Don’t get confused with [2.75] =2 or [n.xyz]=n. For any positive number, the Square Bracket Function (say it Floor Function or Greatest Integer Function) allows you to leave fractional part of the number. Read more at http://en.wikipedia.org/wiki/Floor_and_Ceiling_Functions .
2.Let n be a fixed positive integer. Two integers a and b are said to be congruent modulo n, symbolised by , if n divides the difference a-b. For simple understanding is same to for any being an integer. We can simplify it for as . In particular conditions, ; , . More details at http://mathworld.wolfram.com/Congruence.html . //////////
Well, it’s your turn now:
Find on which weekdays these dates fall:
1. July 4, 1776
2. October 19, 1992
3. August 15, 1947
4. March 21, 1688
5. June 8, 2333.
Applied mathematics is one which is used in day-to-day life, in solving tensions (problems) or in business purposes. Let me write an example:
George had some money. He gave 14 Dollars to Matthew. Now he has 27 dollars. How much money had he?
If you are familiar with day to day calculations —you must say that George had 41 dollars, and since he had 41, gave 14 to Matthew saving 27 dollars. That’s right? Off course! This is a general(layman) approach. ‘How will we achieve it mathematically?’ —We shall restate the above problem as another statement (meaning the same):
some moneydollars. He gave 14 dollars to Matthew. Now he has 27 dollars. How much money he had?Find the value of .
This is equivalent to the problem asked above. I have just replaced ‘some money’ by ‘x dollars’. As ‘some’ senses as unknown quantity— does the same. Now all we need to get the value of x.
When solving for , we should have a plan like this:
|He gave to Matthew||14 dollars|
|Now he must have||dollars|
But problem says that he has 27 dollars left. This implies that dollars are equal to 27 dollars.
contains an alphabet x which we assumed to be unknown—can have any certain value. Statements (like ) containing unknown quantities and an equality are called Equations. The unknown quantities used in equations are called variables, usually represented by bottom letters in english alphabet (e.g.,). Top letters of alphabet (..) are usually used to represent constants (one whose value is known, but not shown).
Now let we concentrate on the problem again. We have the equation x-14=27.
Now adding 14 to both sides of the equal sign:
So, is 41. This means George had 41 dollars. And this answer is equal to the answer we found practically. Solving problems practically are not always possible, specially when complicated problems encountered —we use theory of equations. To solve equations, you need to know only four basic operations viz., Addition, Subtraction, Multiplication and Division; and also about the properties of equality sign.
We could also deal above problem as this way:
-14 transfers to another side, which makes the change in sign of the value, i.e., +14.
When we transport a number from left side to right of the equal sign, the sign of the number changes and vice-versa. As here -14 converts into +14; +18 converts into -18 in example below:
Please note, any number not having a sign before its value is deemed to be positive—e.g., 179 and +179 are the same, in theory of equations.
Before we proceed, why not take another example?
Marry had seven sheep. Marry’s uncle gifted her some more sheep. She has eighteen sheep now. How many sheep did her uncle gift?
First of all, how would you state it as an equation?
or, (just to illustrate that 7=+7)
So, Marry’s uncle gifted her 9 sheep. ///
Now tackle this problem,
Monty had some cricket balls. Graham had double number of balls as compared to Monty. Adam had also 6 cricket balls. They all collected their balls and found that total number of cricket balls was 27. How many balls had Monty and Graham?
As usual our first step to solve this problem must be to restate it as an equation. We do it like this:
Monty had (let) x balls.
Then Graham must had balls.
Adam had 6 balls.
The total sum=
But that is 27 according to our question.
Here multiplication sign converts into division sign, when transferred.
Since , we can say that Monty had 7 balls (instead of x balls) and Graham had 14 (instead of ).
Types of Equations
They are many types of algebraic equations (we suffix ‘Algebraic’ because it includes variables which are part of algebra) depending on their properties. In common we classify them into two main parts:
1. Equations with one variable (univariable algebraic equations, or just Univariables)
2. Equations with more than one variables (multivariable algebraic equations, or just Multivariables)
Equations consisting of only one variable are called univariable equations.
All of the equations we solved above are univariables since they contain only one variable (x). Other examples are:
(e is a constant).
Univariables are further divided into many categories depending upon the degree of the variable. Some most common are:
- Linear Univariables: Equations having the maximum power (degree) of the variable 1.
is a general example of linear equations in one variable, where a, b and c are arbitrary constants.
- Quadratic Equations: Also known as Square Equations, are ones in which the maximum power of the variable is 2.
is a general example of quadratic equations, where a,b,c are constants.
- Cubic Equations: Equations of third degree (maximum power=3) are called Cubic.
A cubic equation is of type ; where a,b,c,d are constants.
- Quartic Equations: Equations of fourth degree are Quartic.
A quartic equation is of type .
Similarly, equation of an n-th degree can be defined if the variable of the equation has maximum power n.
Some equations have more than one variables, as etc. Such equations are termed as Multivariable Equations. Depending on the number of variables present in the equations, multivariable equations can be classified as:
1. Bi-variable Equations - Equations having exactly two variables are called bi-variables.
; ; , where k is constant; etc are equations with two variables.
Bivariable equations can also be divided into many categories, as same as univariables were.
A.Linear Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 1.
For example: is a linear but is not.
B. Second Order Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 2.
For example: , are of second order.
Similarly you can easily define n-th order Bivariable equations.
2. Tri-variable Equations: Equations having exactly three variables are called tri-variable equations.
; ; , where k is constant; etc are trivariables. (Further classification of Trivariables are not necessary, but I hope that you can divide them into more categories as we did above.)
Similary, you can easily define any n-variable equation as an equation in which the number of variables is n.
Out of these equations, we shall discuss only Linear Univariable Equations here (actually we are discussing them). ////
We have already discussed them above, for particular example. Here we’ll discuss them for general cases.
As told earlier, a general example of linear univariable equation is .
We can adjust it by transfering constants to one side and keeping variable to other.
this is the required solution.
Example: Solve .