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Basic Definitions : Group Theory #group theory

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Basic Definitions :  * Binary Operation  * Algebraic Structure * Quasi- Group or Groupoid * Semi Group * Monoid * Group * Abelian Group Binary Operation :            Let S be a non-empty set . If f : SxS→R is a mapping , then f is called binary                   operation or binary  composition on S.           Thus        If a relation in S is such that every pair ( distinct or equal ) of elements of S  taken in definite  order is associated with a unique element of S then it is called a binary operation in S. Otherwise the relationis not binary operation in S and the relation is simply an operation in S.      (a,b) ๐Ÿ„ SxS , ∃ a unique element f(a,b) ๐Ÿ„ S. We observe that addition, multiplication, subtraction are binary operations in R and division is not a binary operation in R  why because division by 0 is not defined...
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Euler's summation formula : Number Theory #Euler's #summation #formula #: #Number #Theory

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Euler's summation formula : Statement :        If f(x) has continuous derivative f’ on [a,b] where 0 < a< b , then          ๐œฎ a<n ≤ b    f(n) = ∫ f(t)dt + ∫ (t – [t] ) dt +f(b)( [b] – b) -f(a) ( [a] -a ).   Proof :    Suppose f(x) has continuous derivative and f’(x) is in the closed interval [a,b] where 0 < a < b.   Let [a] =m and   [b] = k.   Then ๐œฎ a<n ≤ b    f(n) = ๐œฎ m<n ≤ k    f(n) = ๐œฎ f(n)   …(i)            | where in third sigma n is from m+1 to k |   Suppose (n-1) and n are two integers in [a,b] and t lies between (n-1) and n. Then ∫ n-1 n [t] f’(t) dt = ∫ (n-1) f’(t) dt                                ...
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big "Oh" notation : Number Theory #BIG #oh #NOTATION #: #Number #Theory

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big "Oh" notation :     Definition :                         Suppose g(x) > 0 for all real values x ≥ a and f(x) is a real valued function                 such that    f(x)/g(x) is bounded for x ≥ a . Then we say that " f(x) is of large order g(x) " or "  f(x) is of order g(x) ".  In this case , we write f(x) = O(g(x))  ( we read f(x) is big Oh of g(x).  Equivalently , we can say that f(x) = O(g(x))  if there exists M > 0 such that | f(x)/g(x) | ≤ M  for all x ≥ a , or |f(x) | ≤ M |g(x)|  or for all x ≥ a . Note :     i) f(x) = h(x) + O g(x) means that f(x) - h(x) = O g(x)  ⇒ | f(x) - h(x) | ≤ M g(x) for some          M > 0.   ii) Suppose f(t) = O g(t) for t ≥ a ⇒ | f(t) | ≤ M g(t) for t ≥ a and for some M > 0.               ...
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Selberg Identity : Number Theory #Selberg #Identity #: #Number #Theory

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Selberg Identity :     For n ≥ 1 , we have    โดท(n) logn + ๐œฎ โดท(d) โดท(n/d) = ๐œฎ ฮผ(d) log 2 (n/d). Proof :      Suppose n ≥ 1 .   We know that log n = ( โดท * u ) (n)   ⇒ u(n)logn = ( โดท * u ) (n) | since u(n) = 1)   ⇒ u’(n) = ( โดท * u ) (n)    …………… ① By differentiating on both sides , we get   u’’(n) = ( โดท * u )’ (n)              = ( โดท’ * u ) (n) + ( โดท * u’ ) (n)             =  ( โดท’ * u ) (n) + ( โดท *  ( โดท * u ) (n)    | Since from    ① |  u’’(n)  =   ( โดท’ * u ) (n) + ( ( โดท *    โดท)) * u ) (n)                                     ...
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Bracket Function : Number Theory #Bracket #Function #: #Number #Theory

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    Bracket Function :                   The function I : R → Z defined by I(x) = n where n ≤ x < n+1 is called the bracket function or the step function or the integral part function .   Notation :                    Integral part of x in R is denoted by I(x) or [x]. Definition :                     If x ๐žŠ R , x - [x] is called the Fractional part of x.    Note :         [x] ≤ x < [x] + 1  or x-1 < [x] ≤ x.       For every x ๐žŠ R, x ≥ [x] ,  hence the fractional part of x is always non-negative       x ๐žŠ Z ⇔ x=[x].   For example :         * For 14/3 = 4.6666... , [ 14/3] = 4.        * For 3/4, [3/4] = 0   because 0 < 3/4 < 1.        * For ฯ€ = 22/7 , we...
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