DR.M.KOTHANDARAMAN

 ASTRONOMY      Astronomy   is defined as the study of the objects that lie beyond our planet Earth and the processes by which these objects interact with one another. We will see, though, that it is much more. It is also humanity’s attempt to organize what we learn into a clear history of the universe, from the instant of its birth in the Big Bang to the present moment. Throughout this book, we emphasize that science is a   progress report —one that changes constantly as new techniques and instruments allow us to probe the universe more deeply. In considering the history of the universe, we will see again and again that the cosmos  evolves ; it changes in profound ways over long periods of time. For example, the universe made the carbon, the calcium, and the oxygen necessary to construct something as interesting and complicated as you. Today, many billions of years later, the universe has evolved into a more hospitable place for life. Tracing the evol...

complex analaysis  is a branch of mathematics that deals with complex numbers, their functions, and their calculus. In simple terms, complex analysis is an extension of the calculus of real numbers to the complex domain. We will extend the notions of continuity, derivatives, and integrals, familiar from calculus to the case of complex functions of a complex variable. In doing so we will come across analytic functions, which form the centerpiece of this introduction. In fact, to a large extent complex analysis is the study of analytic functions. The basic ingredient of complex analysis is an analytic function, or that we know so well in calculus as a differentiable function. Any  complex number   z  can be thought of as a point in a plane ( x,y ), so  z = x+iy,  where  i=√-1.   In a similar fashion, any complex function of a complex variable  z  can be separated into two functions, as in,   f(z)=u(z)+iv(z),  or,  f(x,y...

  Inclusion-Exclusion and its various Applications Difficulty Level :   Easy Last Updated :   17 Aug, 2021 Read Discuss In the field of Combinatorics, it is a counting method used to compute the cardinality of the union set. According to basic  Inclusion-Exclusion principle :    For 2 finite sets  and  , which are subsets of Universal set, then  and  are disjoint sets.    Hence it can be said that,  . Similarly for 3 finite sets  ,  and  ,    Principle : Inclusion-Exclusion principle says that for any number of finite sets  , Union of the sets is given by = Sum of sizes of all single sets – Sum of all 2-set intersections + Sum of all the 3-set intersections – Sum of all 4-set intersections .. +  Sum of all the i-set intersections.  In general it can be said that,    Properties :    Computes the total number of elements that satisfy at least one of several properti...

  Algebraic Structure A non empty set S is called an algebraic structure w.r.t binary operation (*) if it follows the following axioms:  Closure: (a*b) belongs to S for all a,b ∈ S.  Example:   S = {1,-1} is algebraic structure under *  As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belong to S.  But the above is not an algebraic structure under + as 1+(-1) = 0 not belongs to S.  Semi Group A non-empty set S, (S,*) is called a semigroup if it follows the following axiom:  Closure: (a*b) belongs to S for all a, b ∈ S.   Associativity:  a*(b*c) = (a*b)*c ∀ a, b ,c belongs to S. Note:  A semi-group is always an algebraic structure.  Example:  (Set of integers, +), and (Matrix ,*) are examples of semigroup.   Monoid A non-empty set S, (S,*) is called a monoid if it follows the following axiom:    Closure: (a*b) belongs to S for all a, b ∈ S.  Associativity:  a*(b*c) = (a*b)*c ∀ a, b, c belongs ...