Mathematics Seminar topics for B.sc Students

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Mathematics Seminar topics for B.sc Students
Mathematics Seminar topics for B.sc Students

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Mathematics Seminar topics for B.sc Students
Mathematics Seminar topics for B.sc Students

1.

Mathematics Seminar topics for B.sc Students

 WELCOME
Seminar Presentation On Charpit’s Methods for Solving Non-linear Partial Differential Equations of Order One 

.
                                                                                                 Name :Monoj Boruah                    Roll No:03                                                             Bsc 3rd  Sem. 2020-21                         
            Department of Mathematics

          Presented By
ContentIntroduction Of Partial Differential Equations .About Charpit’s Method.Charpit’s Method Proof ( General Method of solving partial differential equations of Order One but of any degree )Working RoleConditions For Applying Charpit’s MethodExamples of applying Charpit’s Method.Applications Of Charpit’s methodAcknowledgementRefferencesConclusionThank you


Introductions Of Partial Differential Equations. 
About Charpit’s methodA linear partial differential equation of order one with two or more independent variables is solved by Lagrange’s method. But in many problems of science and engineering when we arrive at a non-linear partial differential equation of order one with two or more independent variables then we require new methods of solution. Thus, we describe Charpit’s methods for solving non-linear partial differential equations of order one. This method is used for solving non-linear partial differential equations of order one involving two independent variables, the method for solving f ( x , y ,z, p , q)=0  involving two independent variables x and y is given by Charpit and is known as Charpit’s method.
Charpit’s Method ( General Method of solving partial differential equations of Order One but of any degree ) 
 
 (Mathematics Seminar topics for B.sc Students)
Working Rules of Charpit’s Method for solving Non-Linear partial differential equations of order one with two independent variables .

Step 1: Putting the terms of the equations is left hand side and denote the whole expression by f or F.Step 2: Write Charpit’s auxillary equations.Step 3: Find the values of different partial derivatives of f or F and put in Charpit’s auxillary equations.Step 4: Select two proper factions such that the resulting intregal may come out in the symplest form involving atleast one of p and q .Step 5: Solving these symplest relation (obtain in step 4) with the given equations to find p and q .Step 6: Putting the values of p and q in dz=pdx +qdyStep 7:Intregate last equations we get the required general solutions.  
Conditions for Applying Charpit’s Method    (Mathematics Seminar topics for B.sc Students)
 
 
 
Applications of Charpit’s Method
Charpit’s method is used for solving non-linear partial differential equations of order one involving two independent variables. Mainly This method is used in the many problems of science and engineering when we arrive at a non-linear partial differential equation of order one with two or more independent variables .Also Charpit’s Method is use to analyse the power system’s stability.
Acknowledgement
I would like to express my gratitude and appreciation to all those who gave me the possibility to complete this Seminar . A special thanks to my respected teachers and my dear friends ,who help me to stimulating suggestion and encouragement to complete this.
References  (Mathematics Seminar topics for B.sc Students)
www.wikipedia.orghttps://www.math24.netAdvance Differential Equations(S. Chand)Ordinary And Partial Differential Equations (S.Chand)www.google.com

 
Thank YouMath may not teach me how to add love or substract hate ,but it gives me every reason to hope that every problem has a solutions.

 2.Mathematics Seminar topics for B.sc Students

Mathematics Seminar topics for B.sc Students

  A Seminar Presentation On Symmetries of a Square and dihedral group : A Mathematical Investigation



Presented by Dibakar Hazarika Roll no:54Class: b.sc 3rd semester Mathematics department
Introduction Aim Symmetries of a Squarecayley table Dihedral group Some important question Reference conclusionContent
Symmetry comes from a Greek word meaning ‘to measure together’ and is widely used in the study of geometry. Mathematically, symmetry means that one shape becomes exactly like another when we move it in some way( turn, flip or slide). For two objects to be symmetrical, they must be the same size and shape, with one object having a different orientation from the first. There can also be symmetry in one object, such as a face. If you draw a line of symmetry down the center of your face, you can see that the left side is a mirror image of the right side. Not all objects have symmetry; if an object is not symmetrical, it is called asymmetric. There are three basic types of symmetry: rotational symmetry, reflection symmetry, and point symmetry. Introduction
The aim of this investigation is to describe clearly the symmetries of a square and Dihedral Group ,after that we have to show that the set of symmetries Is a group . Aim
Symmetries of a Square Mathematical Working:   4      3     1     2 Suppose we remove a square region from a plane move it in some possible ways (Rotation And Reflection )such that  the square will look the same after the motion  as before .Consider the square with vertices denoted by 1,2,3 and 4 .  (Mathematics Seminar topics for B.sc Students)
               R0 :    2                 R90 :           Anticlockwise Rotation                  3


1        2                    3


1        2                  3


1        2                   2


4         1Rotation of 00 Rotation of 900 
               R180 :    2                 R270 :           AntiClockwise Rotation                  3


1        2 2                      1


3        4                  3


1        21                        4


2        3Rotation of 1800 Rotation of 2700 
               H :    2                 V :                         Reflections                  3


1        2 1                       2


4        3                  3


1        2                   2


4         1Flip about Horizontal axis Flip about Vertical axis
,Reflections D:D`:Flip about main diagonal Flip about other diagonal                    1


3         22314
Let’s investigate some consequences of the fact that every motion is equal to one of the eight listed in above.Suppose a square is repositioned by a rotation of 900 followed by a flip about the horizontal axis of symmetry





Thus, we see that this pair of motions—taken together—is equal to the single motion D. This observation suggests that we can compose two motions to obtain a single motion. And indeed we can, since the eight motions may be viewed as functions from the square region to itself, and as such we can combine them using function composition.With this in mind, we write H R90 5 D The eight motions R0, R90, R180, R270, H, V, D, and D9, together with the operation composition, form a mathematical system called the dihedral group of *order 8. It is denoted by D4


* (the order of a group is the number of elements it contains).        (Mathematics Seminar topics for B.sc Students)            3


1        2 Rotation of 900 3                       2


4        1 4                       1


3        2 Flip about horizontal 
Now we construct an operation table or cayley table for a square region i.e. d4


To make the table
We can take an example ——————-V * R270  = ?
Since   V* R270 =D
Now we have to Show that how did We obtain D’  (Mathematics Seminar topics for B.sc Students)



                                                                                                            = D                                                                                                           

*R0R90R180R270 HVDD’R0R90R180R270HVDDD’1                        4
          R270
2        34                       1
             
3        2
Similarly we complete the table. It is like that….













         Now we examine that the set of symmetries Is a group or not?1.Clouser:-From the composition table we see that the set of symmetries (D4) Is closed w.r.t. the binary operation *.2.Associativity:-    (R270 * H) * D = R270 * (H * D)                D * D = R270 * R90    R0 = R0 Hence it is clear that set of symmetries (D4) Is associative  w.r.t. the binary operation *.*R0R90R180R270 HVDD’R0R0R90R180R270 HVDD’R90R90R180R270 R0D’DHVR180R180R270 R0R90VHD’DR270R270R0R90R180DD’VHHHD’VDR0R180R90R270 VVDHD’R180R0R270 R90DDVD’HR270 R90R0R180D’D’HDVR90R270 R180R0
3.Idendity : Here R0 is the Idendity element.4.Inverse:- From the composition table we see that the inverse of (R0, R90, R180, R270, H , VD , D’) Are (R0 , R270, R180, R90, H , V , D , D’ ) respectively.Hence the set of symmetries Is a group         Again we test the commutative property to clear that the set of symmetries Is a abelian group or not.For commutative:1. R270 * R180 = H * V                R90  ≠ R1802. D * H =   R180 * R90       R270 ≠ R270 Since it is not commutative hence the set of symmetries Is  not a abelian group.






DIHEDRAL GROUP ( D4 )      It  is a group of symmetries of regular polygon of n sides. This group consist of 2n symmetries, hence order of the group is 2n. Out of 2n ,  n are rotational symmetries and n are reflectional symmetries. 
  The rotational is made around the normal to the plane of polygon at the center and reflection are made about the line dividing the plane  of polygon into two equal parts provided that these lines passes through either vertices or mid point of sides.

             c                                  b                                                          c                                   b



            d                                  a                                                         d                                    a  
 
 Some important question        
2. R120: Rotation of 1200             1            3                2                       3              1               2 3. R240:rotation of 2400                          1                                                       2               2                        3                            3                        1 4.D1 : flip about D1           1                                                          1              2            3                                3                       2 Rotation
D2: Flip about D2    1                                                      3                      2                     3                               2                     1 D3 :flip about D3 1      2                   2 3           1                     3  REFLECTION 
Now we construct an operation table for D3









Now we show that D3  is abelian or not??From the above composition table we see that D3 is closed and associative w r t the binary operation *Here R0 is the identity element.From the above composition table we see that the inverse of (R0,R120,R240,D1,D2,D3)Are (R0,R240,R120,D1,D2,D3) respectively.For commutative D1*R120= R120*D          D2 ≠ D3Hence it is not commutative so D3 is not an abelian group.


*R0R120R240D1D2D3R0R0R120R240D1D2D3R120R120R240R0D3D1D2R240R240R0R120D2D3D1D1D1D2D3R0R120R240D2D2D3D1R240R0R120D3D3D1D2R120R240R0
Reference
Contemporary abstract algebra, J.A. GallianDegree mathematics (group theory -1) G. Hazarika, C.Chutia, K.Gogoi M.J. Borah.Internet  
Conclusion
The elements of the group of the symmetries of a square are the rotations and reflections. Called this set DR0=Rotation of 3600R90 =Rotation of 900R180=Rotation of 1800R270=Rotation of 2700H=Reflect about horrizental axisV=Reflect about vertical axisD=Reflect about main diagonalD`=Reflect about other diagonal
Set D4=(R0, R90 ,R180 ,R270 ,H, ,V, D, D`)Together with a low of composition satisfies the associative law , identity low , and inverse low . The identity element of D4 is R0,  and  every element of set D4 has unique inverse.Therefore, (D4, *) is a group. This group is nonabelian because it did not satiesfies commutative low. 
The only way to learn mathematics is to do mathematics. Thankyou  (Mathematics Seminar topics for B.sc Students)

 Mathematics seminar topics pdf Mathematics Seminar topics for B.sc Students

 
Seminar presentation on an abelian group PRESENTED BYMUNINDRA REGONROLL NO :07B.SC 3rd SEMYEAR: 2020-21


CONTENTS  INTRODUCTIONHISTORYBINARY OPERATIONALGEBRIAC STRUCTURE  DEFINATION OF GROUPDEFINITION OF  ABELIAN GROUP PROPERTIES OF  ABELIAN GROUPEXAMPLE OF  ABELIAN GROUPDEFINITION OF FINITE ABELIAN GROUPDEFINITION OF INFINITE ABELIAN GROUPDEFINITION OF NON ANELIAN GROUPCONCLUSIONREFERENCEACKNOWLEDGEMENT  
Seminar presentation on an abelian group PRESENTED BYMUNINDRA REGONROLL NO :07B.SC 3rd SEMYEAR: 2020-21
 (Mathematics Seminar topics for B.sc Students)

CONTENTS  INTRODUCTIONHISTORYBINARY OPERATIONALGEBRIAC STRUCTURE  DEFINATION OF GROUPDEFINITION OF  ABELIAN GROUP PROPERTIES OF  ABELIAN GROUPEXAMPLE OF  ABELIAN GROUPDEFINITION OF FINITE ABELIAN GROUPDEFINITION OF INFINITE ABELIAN GROUPDEFINITION OF NON ANELIAN GROUPCONCLUSIONREFERENCEACKNOWLEDGEMENT


 (Mathematics Seminar topics for B.sc Students)

INTRODUCTION  In a mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups.

HISTORY    Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlines many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras.
Binary operation   Let G be a set. A binary operation on G is a function that assigns each order pair of elements of G an elements of G.              f : G×G → G  Remark :         * is a binary operation on G iff a*b €G .
Algebraic Structure:  A non empty set is together with one or more then one binary operation is called algebraic structure.  Example : – (R,+∙) is an algebraic structure. (N,+),(Z,+),(Q,+) are algebraic structure. 

 (Mathematics Seminar topics for B.sc Students)



Definition of Group:    An algebraic structure {G,*} where G is a non empty set with the binary operations ‘*’ defined on it, is said to be a group is the following actions satisfied.Closure property:   G is closed under the operations ‘*’ i . e.  a * b€ G for all of a , b€ G.Associative:  The binary operation ‘*’ is associative  i.e. a ,b , c € G, then (a * b)*c=a*(b * c)Identity: There is an elements e (called the identity) in G such that for all x€G, e * a=a * e=x, for all of a € GInverse : For every a € G, there is an elements a΄ in G, such that a * a΄=a * a΄=e






















DEFINITION OF AN ABELIAN GROUP:     The goal  this section is to look at several properties of an abelian groups and see how they relate to general properties of modules of less independent of the previous one. We first need to write down the definition of a group. To shorten it a bit, we use the definition that a binary operation is one which the closure property holds. Definition: The group (G,*) is abelian if its binary operation is commutative .   i. e: a*b=b*a, for all a , b €G.  (Mathematics Seminar topics for B.sc Students)


Properties of abelian groups:    Abelian Group:   Let G be a non empty set and * be a binary operation defined on it, then the structure ( G,*) is said to be a group, if the following axioms or properties are satisfied.Closure: G is close under the operation *  i. e a *b €G for all a,b €G. Associative:  For all a , b, c €G, we have (a*b)*c=a*(b*c)Identity : There is an element e (e is called the identity) in G such that for all x€G, e*x=x*e=x.Inverses : Corresponding to each a €G, there is an element a׳€G (a′ is called an inverse of a) such that a*a′=a′*a=e Commutative :  A group (G,*) is an abelian if its binary operation commutative i.e a*b=b*a for all a,b € G.


Example of abelian group: 
     Identity:           Let e be the identity of Q+ let for all a € Q+  there for e*a=a             e*a/2=a                    e=2  There for 2€Q+ is the identity element.  Inverse: Let b € Q+ be the inverse element of a € Q+.     a*b=e    a b/2=2        b=4/a €Q+ Hence 4/a is the inverse of a.Commutative :  Let a, b €Q+ then   a*b=e   a*b= a b/2=b a/2=b*a  Hence (Q+,*) is an abelian.         (Mathematics Seminar topics for B.sc Students)
     
  Example:2. Show that the subset {1,-1, i, -i} of the complex is an abelian group under complex multiplication. Solution:  Now we form the composition table  




   ∙1-1i-i1
1
-1
i
-i
-1
-1
1
-i
i
i
i
-i
-1
1
-i
-i
i
1
-1

  Closure property:  From the composition we see that the given set is close  under multiplication.
 Associative:    The multiplication of the complex number is associative.     (Mathematics Seminar topics for B.sc Students)
 Existence of identity: Here 1 is the identity element of G  Inverses:    From the composition table we see that the inverse 0f 1,-1,i,-i are 1,-1,-i,i respectively. Commutative :   The multiplication of complex numbers is commutative.Hence  {1,-1,i,-i} is an abelian group with respect to multiplication.
  

   Finite abelian group:      A finite abelian group is a group satisfying the following conditions: It is both finite and abelian .It is isomorphic to a direct product of finitely many finite cyclic groups.It is isomorphic to a direct product of abelian groups of prime number order.It is isomorphic to a direct product of cyclic groups of prime power order.   Example : Obviously, all finite groups are finitely generated. For example , the group s3 is generated by the permutations (12) and (123). The group  Z×Zn is an infinite group but is finitely generated by {(1,0),(0,1)}.

 Definition of Infinite abelian group    The simplest infinite abelian group is the infinite cyclic group  Z. Any finitely generated abelian group A is isomorphic to  the direct sum of r copies of Z and a finite abelian group, which in terms is decomposable into  a direct sum of finitely many cyclic groups of prime power orders. Even through the decomposition is not a unique, the number  r, called the rank of A, and the prime powers giving the orders of finite orders of finite cyclic summands are uniquely determined. There are also examples of infinite bounded abelian groups, such as (Z,Zn) N for some positive integers n.   Every infinite abelian group has a at least one element of infinite order. Some example of infinite abelian groups :       (Z,+), (Q,+), (R,+), (C,+),    (Mathematics Seminar topics for B.sc Students)
  Non – Abelian group:     In mathematics , and specifically in group theory, a non abelian group , sometimes called a non commutative group, is a group (G,*) in which there exists at least one pair of elements a and b of ,G such that a * b ≠ b * a.    Non abelian groups are pervasive in mathematics and physics. One of simplest example of  non Abelian group is the dihedral group D3 of order 6 . It is the simplest finite non- abelian group. A common group physics is the rotation group s3 in three dimensions.
Application of an abelian group :     Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors.
  
    

  Conclusion:       The concept of an abelian group underlies many fundamental algebraic structure, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non abelian counterparts, and finite abelian groups are very well understood and fully classified.  
    Reference:           I am collect data  Degree Mathematics (Group Theory – I)Abstract Algebra, seven Edition (John B. Fraleigh )Internet   (Mathematics Seminar topics for B.sc Students)

ACKNOWLEDGEMENTFirst I express gratitude my teachers ,who  by their  encouragement interest and advice helped me to write this seminar .



             THANK   YOU





INTRODUCTION  In a mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups.  (Mathematics Seminar topics for B.sc Students)

HISTORY    Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlines many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras.
Binary operation   Let G be a set. A binary operation on G is a function that assigns each order pair of elements of G an elements of G.              f : G×G → G  Remark :         * is a binary operation on G iff a*b €G .
Algebraic Structure:  A non empty set is together with one or more then one binary operation is called algebraic structure.  Example : – (R,+∙) is an algebraic structure. (N,+),(Z,+),(Q,+) are algebraic structure. 





Definition of Group:    An algebraic structure {G,*} where G is a non empty set with the binary operations ‘*’ defined on it, is said to be a group is the following actions satisfied.Closure property:   G is closed under the operations ‘*’ i . e.  a * b€ G for all of a , b€ G.Associative:  The binary operation ‘*’ is associative  i.e. a ,b , c € G, then (a * b)*c=a*(b * c)Identity: There is an elements e (called the identity) in G such that for all x€G, e * a=a * e=x, for all of a € GInverse : For every a € G, there is an elements a΄ in G, such that a * a΄=a * a΄=e  (Mathematics Seminar topics for B.sc Students)






















DEFINITION OF AN ABELIAN GROUP:     The goal  this section is to look at several properties of an abelian groups and see how they relate to general properties of modules of less independent of the previous one. We first need to write down the definition of a group. To shorten it a bit, we use the definition that a binary operation is one which the closure property holds. Definition: The group (G,*) is abelian if its binary operation is commutative .   i. e: a*b=b*a, for all a , b €G.
 (Mathematics Seminar topics for B.sc Students)

Properties of abelian groups:    Abelian Group:   Let G be a non empty set and * be a binary operation defined on it, then the structure ( G,*) is said to be a group, if the following axioms or properties are satisfied.Closure: G is close under the operation *  i. e a *b €G for all a,b €G. Associative:  For all a , b, c €G, we have (a*b)*c=a*(b*c)Identity : There is an element e (e is called the identity) in G such that for all x€G, e*x=x*e=x.Inverses : Corresponding to each a €G, there is an element a׳€G (a′ is called an inverse of a) such that a*a′=a′*a=e Commutative :  A group (G,*) is an abelian if its binary operation commutative i.e a*b=b*a for all a,b € G.  (Mathematics Seminar topics for B.sc Students)


Example of abelian group: 
     Identity:           Let e be the identity of Q+ let for all a € Q+  there for e*a=a             e*a/2=a                    e=2  There for 2€Q+ is the identity element.  Inverse: Let b € Q+ be the inverse element of a € Q+.     a*b=e    a b/2=2        b=4/a €Q+ Hence 4/a is the inverse of a.Commutative :  Let a, b €Q+ then   a*b=e   a*b= a b/2=b a/2=b*a  Hence (Q+,*) is an abelian.       
     
  Example:2. Show that the subset {1,-1, i, -i} of the complex is an abelian group under complex multiplication. Solution:  Now we form the composition table  
 (Mathematics Seminar topics for B.sc Students)



   ∙1-1i-i1
1
-1
i
-i
-1
-1
1
-i
i
i
i
-i
-1
1
-i
-i
i
1
-1

  Closure property:  From the composition we see that the given set is close  under multiplication.
 Associative:    The multiplication of the complex number is associative.    
 Existence of identity: Here 1 is the identity element of G  Inverses:    From the composition table we see that the inverse 0f 1,-1,i,-i are 1,-1,-i,i respectively. Commutative :   The multiplication of complex numbers is commutative.Hence  {1,-1,i,-i} is an abelian group with respect to multiplication.  (Mathematics Seminar topics for B.sc Students)
  

   Finite abelian group:      A finite abelian group is a group satisfying the following conditions: It is both finite and abelian .It is isomorphic to a direct product of finitely many finite cyclic groups.It is isomorphic to a direct product of abelian groups of prime number order.It is isomorphic to a direct product of cyclic groups of prime power order.   Example : Obviously, all finite groups are finitely generated. For example , the group s3 is generated by the permutations (12) and (123). The group  Z×Zn is an infinite group but is finitely generated by {(1,0),(0,1)}.  (Mathematics Seminar topics for B.sc Students)

 Definition of Infinite abelian group    The simplest infinite abelian group is the infinite cyclic group  Z. Any finitely generated abelian group A is isomorphic to  the direct sum of r copies of Z and a finite abelian group, which in terms is decomposable into  a direct sum of finitely many cyclic groups of prime power orders. Even through the decomposition is not a unique, the number  r, called the rank of A, and the prime powers giving the orders of finite orders of finite cyclic summands are uniquely determined. There are also examples of infinite bounded abelian groups, such as (Z,Zn) N for some positive integers n.   Every infinite abelian group has a at least one element of infinite order. Some example of infinite abelian groups :       (Z,+), (Q,+), (R,+), (C,+),  
  Non – Abelian group:     In mathematics , and specifically in group theory, a non abelian group , sometimes called a non commutative group, is a group (G,*) in which there exists at least one pair of elements a and b of ,G such that a * b ≠ b * a.    Non abelian groups are pervasive in mathematics and physics. One of simplest example of  non Abelian group is the dihedral group D3 of order 6 . It is the simplest finite non- abelian group. A common group physics is the rotation group s3 in three dimensions.
Application of an abelian group :     Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors.
    (Mathematics Seminar topics for B.sc Students)
    

  Conclusion:       The concept of an abelian group underlies many fundamental algebraic structure, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non abelian counterparts, and finite abelian groups are very well understood and fully classified.  
    Reference:           I am collect data  Degree Mathematics (Group Theory – I)Abstract Algebra, seven Edition (John B. Fraleigh )Internet 

ACKNOWLEDGEMENTFirst I express gratitude my teachers ,who  by their  encouragement interest and advice helped me to write this seminar .
 (Mathematics Seminar topics for B.sc Students)


             THANK   YOU
 
Seminar presentation on an abelian group PRESENTED BYMUNINDRA REGONROLL NO :07B.SC 3rd SEMYEAR: 2020-21


CONTENTS  INTRODUCTIONHISTORYBINARY OPERATIONALGEBRIAC STRUCTURE  DEFINATION OF GROUPDEFINITION OF  ABELIAN GROUP PROPERTIES OF  ABELIAN GROUPEXAMPLE OF  ABELIAN GROUPDEFINITION OF FINITE ABELIAN GROUPDEFINITION OF INFINITE ABELIAN GROUPDEFINITION OF NON ANELIAN GROUPCONCLUSIONREFERENCEACKNOWLEDGEMENT




INTRODUCTION  In a mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups.

HISTORY    Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlines many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras.
Binary operation   Let G be a set. A binary operation on G is a function that assigns each order pair of elements of G an elements of G.              f : G×G → G  Remark :         * is a binary operation on G iff a*b €G .
Algebraic Structure:  A non empty set is together with one or more then one binary operation is called algebraic structure.  Example : – (R,+∙) is an algebraic structure. (N,+),(Z,+),(Q,+) are algebraic structure. (Mathematics Seminar topics for B.sc Students)





Definition of Group:    An algebraic structure {G,*} where G is a non empty set with the binary operations ‘*’ defined on it, is said to be a group is the following actions satisfied.Closure property:   G is closed under the operations ‘*’ i . e.  a * b€ G for all of a , b€ G.Associative:  The binary operation ‘*’ is associative  i.e. a ,b , c € G, then (a * b)*c=a*(b * c)Identity: There is an elements e (called the identity) in G such that for all x€G, e * a=a * e=x, for all of a € GInverse : For every a € G, there is an elements a΄ in G, such that a * a΄=a * a΄=e(Mathematics Seminar topics for B.sc Students)


DEFINITION OF AN ABELIAN GROUP:     The goal  this section is to look at several properties of an abelian groups and see how they relate to general properties of modules of less independent of the previous one. We first need to write down the definition of a group. To shorten it a bit, we use the definition that a binary operation is one which the closure property holds. Definition: The group (G,*) is abelian if its binary operation is commutative .   i. e: a*b=b*a, for all a , b €G.


Properties of abelian groups:    Abelian Group:   Let G be a non empty set and * be a binary operation defined on it, then the structure ( G,*) is said to be a group, if the following axioms or properties are satisfied.Closure: G is close under the operation *  i. e a *b €G for all a,b €G. Associative:  For all a , b, c €G, we have (a*b)*c=a*(b*c)Identity : There is an element e (e is called the identity) in G such that for all x€G, e*x=x*e=x.Inverses : Corresponding to each a €G, there is an element a׳€G (a′ is called an inverse of a) such that a*a′=a′*a=e Commutative :  A group (G,*) is an abelian if its binary operation commutative i.e a*b=b*a for all a,b € G.(Mathematics Seminar topics for B.sc Students)


Example of abelian group: 
     Identity:           Let e be the identity of Q+ let for all a € Q+  there for e*a=a             e*a/2=a                    e=2  There for 2€Q+ is the identity element.  Inverse: Let b € Q+ be the inverse element of a € Q+.     a*b=e    a b/2=2        b=4/a €Q+ Hence 4/a is the inverse of a.Commutative :  Let a, b €Q+ then   a*b=e   a*b= a b/2=b a/2=b*a  Hence (Q+,*) is an abelian.       
     
  Example:2. Show that the subset {1,-1, i, -i} of the complex is an abelian group under complex multiplication. Solution:  Now we form the composition table  (Mathematics Seminar topics for B.sc Students)




   ∙1-1i-i1
1
-1
i
-i
-1
-1
1
-i
i
i
i
-i
-1
1
-i
-i
i
1
-1

  Closure property:  From the composition we see that the given set is close  under multiplication.
 Associative:    The multiplication of the complex number is associative.    
 Existence of identity: Here 1 is the identity element of G  Inverses:    From the composition table we see that the inverse 0f 1,-1,i,-i are 1,-1,-i,i respectively. Commutative :   The multiplication of complex numbers is commutative.Hence  {1,-1,i,-i} is an abelian group with respect to multiplication.(Mathematics Seminar topics for B.sc Students)
  

   Finite abelian group:      A finite abelian group is a group satisfying the following conditions: It is both finite and abelian .It is isomorphic to a direct product of finitely many finite cyclic groups.It is isomorphic to a direct product of abelian groups of prime number order.It is isomorphic to a direct product of cyclic groups of prime power order.   Example : Obviously, all finite groups are finitely generated. For example , the group s3 is generated by the permutations (12) and (123). The group  Z×Zn is an infinite group but is finitely generated by {(1,0),(0,1)}.

 Definition of Infinite abelian group    The simplest infinite abelian group is the infinite cyclic group  Z. Any finitely generated abelian group A is isomorphic to  the direct sum of r copies of Z and a finite abelian group, which in terms is decomposable into  a direct sum of finitely many cyclic groups of prime power orders. Even through the decomposition is not a unique, the number  r, called the rank of A, and the prime powers giving the orders of finite orders of finite cyclic summands are uniquely determined. There are also examples of infinite bounded abelian groups, such as (Z,Zn) N for some positive integers n.   Every infinite abelian group has a at least one element of infinite order. Some example of infinite abelian groups :       (Z,+), (Q,+), (R,+), (C,+),  
  Non – Abelian group:     In mathematics , and specifically in group theory, a non abelian group , sometimes called a non commutative group, is a group (G,*) in which there exists at least one pair of elements a and b of ,G such that a * b ≠ b * a.    Non abelian groups are pervasive in mathematics and physics. One of simplest example of  non Abelian group is the dihedral group D3 of order 6 . It is the simplest finite non- abelian group. A common group physics is the rotation group s3 in three dimensions.
Application of an abelian group :     Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors.(Mathematics Seminar topics for B.sc Students)
  
    

  Conclusion:       The concept of an abelian group underlies many fundamental algebraic structure, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non abelian counterparts, and finite abelian groups are very well understood and fully classified.  
    Reference:           I am collect data  Degree Mathematics (Group Theory – I)Abstract Algebra, seven Edition (John B. Fraleigh )Internet 

ACKNOWLEDGEMENTFirst I express gratitude my teachers ,who  by their  encouragement interest and advice helped me to write this seminar .(Mathematics Seminar topics for B.sc Students)



             THANK   YOU

 3.Mathematics Seminar topics for B.sc Students

Mathematics Seminar topics for B.sc Students

 1WELCOME
A Seminar Presentation On “Rolle’s Theorem” Name :Gautam Boruah Roll No. : 10BSc 3rd semesterDepartment of Mathematics2020-21 YearSUBMITTED BYDATE:27/01/20212
Introduction HistoryStatement And ProofGeometrical  InterpretationExamplesApplication of Rolle’s theoremConclusionBibliography Acknowledgement(Mathematics Seminar topics for B.sc Students)


3CONTENTS
IntroductionIn Calculus Rolle’s Theorem or Rolle’s Lemma essentially states that any real valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them – that is, a point where the first derivative [the slope of the tangent line to the graph of the function] is zero. 4
History
A version of the theorem was first stated by the Indian Astronomer Bhaskara in the 12th century .The first known formal proof was offered by Michel Rolle in 1691, which used the methods of the differential calculus.55
Statement And ProofThe statement of the theorem :        Suppose that f is a function –        (1) Continuous in the closed interval [a,b] ;        (2) Differentiable on the open interval ]a,b[ ;and (3) f(a)=f(b)         then there exist at least one point c where a<c<b or cϵ]a,b[  such that , fʹ(c)=06
    C+hC-hFirst let f(x) be a constant function .Therefore we have fʹ(x)=0 , for all xϵ]a,b[ So the theorem is established .              Again either f(x) is increasing or decreasing function.  Let f(x) be an increasing function , Now by the theorem we have f(a)=f(b) .So it cease to be increased and begin to decrease at some point cϵ]a,b[ 07Proof:7
 Thus at c we have –          

or  


or 

or


                8
Which contradicts the condition of the theorem, i.e  f(x) is differentiable at all points of the open interval ]a,b[    Therefore we must have –                                                                                                          where cϵ]a,b[                                                                 i.e                             where cϵ]a,b[  or a<c<b    9
     Geometrical InterpretationLet y=f(x) be a curve which is continuous in [a,b] and differentiable in]a,b[ and f(a)=f(b) .  The theorem simplify that between two points with equal ordinates on the graph of f there exist at least one point where the tangent is parallel to X axis 10
     Examples Of Rolle’s TheoremVerify Rolle’s theorem for f(x)=x2 , xϵ]-1,1[   soln : Here the given function is f(x)=x2 where xϵ]-1,1[                Now  (1) f(x) is continuous in [-1,1]                       (2) f(x) is differentiable in ]-1,1[                       (3) f(-1)=f(1)Thus all the three conditions of Rolle’s theorem are satisfied and according to the theorem there exist at least one point c where -1<c<1 such that  fʹ(c)=0 Here we see that fʹ(x)=2x , and fʹ(0)=0 where -1<0<1 .Thus Rolle’s theorem verified .       
11
2.Discuss the applicability of Rolle’s theorem when                        ,
Soln : Here         i.e   f(x)=x    when x>0                      =0    when x=0                      =-x   when x<0  Now f(0+0)12
  And f(0-0)



      Again f(0)=0      Since  f(0+0)=f(0-0)=f(0)=0Thus                     is continuous  Now Rfʹ(0)13
And   Lfʹ(0)


i.e   Thus we see that           (a) f(x) is continuous at x=0 ; where 0ϵ[-1,1]          (b) f(x) is not differentiable at x=0 ; where 0ϵ]-1,1[and   (c) f(-1)=1=f(1)  Thus the Rolle’s theorem is not applicable .14
Application of Rolle’s TheoremRolle’s theorem is often applied with motion problems  such as throwing a ball into the air .It is used for analysing graphs of a company’s yearly performance .15
              Conclusion Rolle’s theorem conclude that if a curve is continuous between two points x=a and x=b, a tangent can be drawn at each and every point between x=a and x=b and functional values at x=a and x=b are equal, then there must be atleast one point between the two points x=a and x=b at which the tangent to the curve is parallel to the x axis16
BibliographyFrom Wikipedia on 25th  january 2021Differential Calculus (B. C. Das l B. N. Mukherjee)
17
Acknowledgement I would like to express my gratitude and appreciation to all those who gave me the possibility to complete this seminar . A special thanks to my respected teachers and my friends , who help to stimulating suggestion and encouragement to complete this .18(Mathematics Seminar topics for B.sc Students)
                                     19      Thank You

 Mathematics seminar topics pdf

4.

 WELCOME
 Presented by                                    

                                                Name :Manoj Boruah                                               Roll No:56                                                                          Bsc 1st Sem. 2019                                                                         Department of Mathematics
SEMINAR   PRESENTATION ON    REDUCTION     FORMULA    for   trigonometry functions. 2
                               contents IntroductionIntregation by partsIntregals involving trigonometric functionsReduction formula for                    andReduction formula for                 andReduction formula for                      and Walli’s   sine   and   cosine   formula. Reduction Formula for Advantages of reduction formula.AcknowledgementReferencesConclusionThank you(Mathematics Seminar topics for B.sc Students)

            Introduction A reduction formula is one that enables us to solve an intregal problem by reducing it to a problem of solving an easier intregal problem ,and then reducing that to the problem of solving an easier problem,and so on.A reduction formula for a given intregal is an intregal which is of the same type as the given intregal but of a lower degree (or order).The reduction formula is used when the given intregal cannot be evaluated otherwise .The repeated application of the reduction formula helps us to evaluate the given intregal . A formula connecting an intregal with a parameter to a similar intregal with a lower value of the parameter is called REDUCTION FORMULA.
             Intregation by parts     Intregation by parts can be used to derive reduction formulas for intregals.     This method of intregation is applied when the intregrand is of the form of product of two functions.Let u and v be two differentiable functions of x .Then applying formula for differential of a product , we have (Mathematics Seminar topics for B.sc Students)



Intregation both sides with respect to x, we get                                                                                                           …..(1) If we substitude u=f(x)and v=                       in (1),we get
                                                                                                          …….(2)
Formula (2) is called “Formula of intregation by parts”


i.e.,intregal of the product of two function = First function ×Intregal of the second function –Intregal of (the derivatives of the first × Intregal of the second function).Instead of (2) , we can use 

Since,

     Intregal involving trigonometric functions  ⮚ Reduction formula for                  and                 .Let (applying intregation by parts) 
Similarly we can prove that

example
⮚Reduction formula for                and Similarly 


Let In=
■Examples of              andEvaluate 
⮚Reduction formula for                     and
Similarly Example:
                        Walli’s   sine   and   cosine   formula.When  n is even and



(Mathematics Seminar topics for B.sc Students)
When n is odd and
                  ,n is even                 ,n is odd
  Reduction Formula for 1.                =
or


EXAMPLE:Solution: hence m=4,n=6since  

Advantages of reduction formula⮚They don’t have any accuracy benefits, but when you would normally need to intregate by parts several times a reduction formula can save you that time .Additionally ,most reduction formulas are found through intregation by parts, so in some sense if you redo that intregation by parts when you don’t need to you are doing unnecessary work.
⮚Most reduction formula are actually just derived from intregation by parts ,so it just saves you time doing it manually and if the answers are both explicit then the accuracy is the same.In fact ,using a reduction formula you are probably less likely to make a mistake by not carrying a minus sign etc.
acknowledgementFirst I expresses gratitude my teachers , who by their encouragement ,interest and advice helped me to write this seminar.
      referenceswww.wikipedia.orghttps://www.math24.netB.Sc. Mathematics (calculus I, CBCS Syllabus)

                  conclusionReduction formula are the formulas used to reduced the quantities ,such as powers ,percent ,logarithms,and so.Reduction formulas are nothing  but reducing an intregal to another intregal involving some variables such as x,n ,t and so on.This reduction can be obtained by using any of the method like intregal by parts ,integration by substitution, and so on. In general recursion or reduction formulas of integration can be denoted by using the letter ‘I’. 
Thank you(Mathematics Seminar topics for B.sc Students)

 1.

Mathematics Seminar topics for B.sc Students

 WELCOME
SEMINAR PRESENTATION TOPIC:- FUNCTION PRESENTED BY:-Dibakar HazarikaB.Sc. 1st-semester rolls No:-40Year:- 2019-20
Contents
(Mathematics Seminar topics for B.sc Students)
INTRODUCTION HISTORY OF FUNCTIONDEFINITIONDIFFERENT TYPES OF FUNCTIONS APPLICATION OF FUNCTIONCONCLUSIONREFERANCE

1. introduction to mathematics, a function is a relation between a set of inputs and a set of permissible outputs. Functions have the property that each input is related to exactly one output

For example, in the function f(x) = x2, f(x) = x2 any input for x will give one output only. Let f(x) = x2 and x=−3, then:f (−3)=(−3)2 =9In the example above, the argument is x=−3 and the output is 9. We write the function as f(−3)=9.

2. History of functions idea of a function was developed in the seventeenth century. During this time, Rene Descartes (1596-1650), in his book Geometry (1637), used the concept to describe many mathematical relationships. The term “function” was introduced by Gottfried Wilhelm Leibniz (1646-1716) almost fifty years after the publication of Geometry. The idea of a function was further formalized by Leonhard Euler (pronounced “oiler” 1707-1783) who introduced the notation for a function, y = f(x).
3. DEFINITION 1
23
4
SOME RELATED BASIC DEFINITIONS 123412345678
4. DIFFERENT TYPES OF FUNCTIONS  1234aeiouAn onto function12 34 5AEIoA function that is not onto
2. one-to-one(1-1) or injective function 12345a.b.c.d.A one-to-one function12345aeioA function that is not one-to-one
one to one and on to function o  Bijective function A function is bijective (one-to-one and onto or one-to-one correspondence) if every element of the codomain is mapped to by exactly one element of the domain. (That is, the function is both injective and surjective.)f:A           B is called bijective if f is both injective and surjective i.e. one to one and on to 
(Mathematics Seminar topics for B.sc Students)
4. Identity function  
  5. FLOUR ANS CEILING FUNCTION floor Ceiling Fractional part 22202.4230.42.9230.9−2.7−3−20.3−2−2−20
5. APPLICATION OF FUNCTION  
 (Mathematics Seminar topics for B.sc Students)
6. CONCLUSION In mathematics, a function is a relationship between two variables called the independent variable and the dependent variable. The dependent variable has at most one value for any specific value of the independent variable. A function is usually symbolized by a lowercase, italicized letter of the alphabet, followed by the independent variable in parentheses. For example, the expression y = f ( x ), read ” y equals f of x,” means that a dependent variable y is a function of the independent variable x. Functions are often graphed, and they usually appear as lines or curves on a coordinate plane.
7. REFERENCE I am collecting the data from          1.B.Sc. Mathematics(D.K. Dalai)          2. Internet
THANK YOU(Mathematics Seminar topics for B.sc Students)

 


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