1.??????(Introduction to Abstract Algebra) - 2.????(Tensor Analysis) - 3.??????(Orthogonal Function Expansion) - 4.????(Green's Function) - 5.???(Calculus of Variation) - 6.????(Perturbation Theory)

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1.??????(Introduction to Abstract
Algebra)-2.????(Tensor Analysis)-3.??????(Orth
ogonal Function Expansion)-4.????(Green's
Function)-5.???(Calculus of Variation)-6.????(
Perturbation Theory)
N968200 ??????
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• Reference
• Birkhoff, G., MacLane, S., A Survey of Modern
  Algebra, 2nd ed, The Macmillan Co, New York,
  1975.
• ???, ????-????, ????, 1989.
• Arangno, D. C., Schaums Outline of Theory and
  Problems of Abstract Algebra, McGraw-Hill Inc,
  1999.
• Deskins, W. E., Abstract Algebra, The Macmillan
  Co, New York, 1964.
• ONan, M., Enderton, H., Linear Algebra, 3rd ed,
  Harcourt Brace Jovanovich Inc, 1990.
• Hoffman, K., Kunze, R., Linear Algebra, 2nd ed,
  The Southeast Book Co, New Jersey, 1971.
• McCoy, N. H., Fundamentals of Abstract Algebra,
  expanded version, Allyn Bacon Inc, Boston,
  1972.
• Hildebrand, F. B., Methods of Applied
  Mathematics, 2nd ed, Prentice-Hall Inc, New
  Jersey, 1972..
• Burton, D. M., An Introduction to Abstract
  Mathematical Systems, Addison-Wesley,
  Massachusetts, 1965.

3
from Wikipedia
David Hilbert
Born January 23, 1862 Wehlau, East Prussia Died
February 14, 1943 Göttingen, Germany Residence
Germany Nationality German Field
Mathematician Erdos Number 4 Institution
University of Königsberg and Göttingen
University Alma Mater University of
Königsberg Doctoral Advisor Ferdinand von
Lindemann Doctoral Students Otto Blumenthal
Richard Courant
Max Dehn
Erich Hecke
Hellmuth Kneser Robert König
Erhard Schmidt Hugo
Steinhaus Emanuel Lasker
Hermann Weyl Ernst Zermelo Known
for Hilbert's basis theorem Hilbert's
axioms Hilbert's problems Hilbert's
program Einstein-Hilbert action Hilbert
space Societies Foreign member of the Royal
Society Spouse Käthe Jerosch (1864-1945, m.
1892) Children Franz Hilbert (1893-1969) Handedne
ss Right handed
The finiteness theorem Axiomatization of
geometry The 23 Problems Formalism
4
Philip M. Morse
Founding ORSA President (1952) B.S. Physics,
1926, Case Institute Ph.D. Physics, 1929,
Princeton University. Faculty member at MIT,
1931-1969.
Methods of Operations Research Queues,
Inventories, and Maintenance Library
Effectiveness Quantum Mechanics Methods of
Theoretical Physics Vibration and
Sound Theoretical Acoustics Thermal
Physics Handbook of Mathematical Functions, with
Formulas, Graphs, and Mathematical Tables
Operations research is an applied science
utilizing all known scientific techniques as
tools in solving a specific problem.
5
Francis B. Hildebrand
George Arfken
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Introduction to Abstract Algebra ??????

• Preliminary notions
• Systems with a single operation
• Mathematical systems with two operations
• Matrix theory an algebraic view

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?? Groupoid

• A goupoid must satisfy
• is closed under the rule of combination
  

9

• Ex. Consider the operation defined on the set
  S 1,2,3 by the operation table below.
• From the table, we see
• 2 (1 3)2 32 but (2 1)
  33 31
• The associative law fails to hold in this
  groupoid(S, )

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??Semigroup

• A semigroup is a groupoid whose operation
  satisfies the associative law.
• (groupoid)

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• Ex. If the operation is defined on by a
  b max a, b ,that is a b is the larger of
  the elements a and b, or either one if ab.
• a (b c) max a, b, c (a b) c
• that shows to be a semigroup
• If and is a
  semigroup, then
• proof.

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?? Monoid

• A semigroup having an identity element
  for the operation is called a monoid.
• 
  (groupoid)
• 
  (semigroup)

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• Ex. Both the semigroups and
  are instances of monoids
• for each
• The empty set is the identity element for the
  union operation.
• for each
• The universal set is the identity element for
  the intersection operation.

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? Group

• A monoid which each element of has
  an inverse is called a group
• 
  (groupoid)
• 
  (semigroup)
• 
  (monoid)

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• If is a group and ,then
• Proof. all we need to show is that
• from the uniqueness of the inverse of
• we would conclude
• a similar argument establishes that

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Commutative ????
Commutative groupoid
Commutative semigroup
Commutative monoid
Commutative group
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• Ex. consider the set of number
  and the operation of ordinary
  multiplication, and Z represents integer.
• Closure
• Associate property
• Identity element
• Commutative property
• is a commutative monoid.

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Ring ?

• A ring is a nonempty set with two
  binary operations and on such that
• is a commutative group
• is a semigroup
• The two operations are related by the
  distributive laws

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• A ring consists of a nonempty set
  and two operations, called addition and
  multiplication and denoted by and ,
  respectively, satisfying the requirements
• R is closed under addition
• Commutative
• Associative
• Identity element 0
• Inverse
• R is closed under multiplication
• Associate
• Distributive law

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Monoid Ring ???

• A monoid ring is a ring with identity
  that is a semigroup with identity

Ring
Monoid ring
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• Ring with commutative property

Commutative
Commutative monoid Ring
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Subring ??

• The triple is a subring of the ring
  
• is a nonempty subset of
• is a subgroup of
• is closed under multiplication

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• The minimal set of conditions for determining
  subrings Let be ring and
  Then the triple is a
  subring of if and only if
• Closed under differences
• Closed under multiplication
• Ex. Let then
  is a subring of
  , since
• This shows that is closed under both
  differences and products.

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Field ?

• A field is a commutative monoid ring
  in which each nonzero element has an inverse
  under

Definition of Field
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Vector ??

• An n-component, or n-dimensional, vector is an n
  tuple of real numbers written either in a row or
  in a column.
• Row vector
• Column vector
• called the components of the
  vector
• n is the dimension of the vector

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Vector space ????

• A vector space( or linear space)
  over the field F consists of the
  following
• A commutative group whose elements are
  called vectors.

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1. A field whose elements are called
   scalars.

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1. An operation ? of scalar multiplication
   connecting the group and field which satisfies
   the properties

V is closed under left multiplication by scalars
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Ex
? Vector Space
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Subspace ?????

• Let V(F) be a vector space over the field F

W(F) is a subspace of V(F)
The minimum conditions that W(F) must satisfy to
be a subspace are
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• If V(F) and V(F) are vector spaces over the
  same field, then the
• mapping f V ? V is said to be
  operation-preserving if

f preserves
V(F) and V(F) are algebraically equivalent
whenever there exists a one-to-one
operation-preserving function from V onto V
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Linear Transformations ????

• Let V and W be vector spaces. A linear
  transformation from V into
• W is a function T from the set V into W with
  the following two
• properties

• x

• T(x)

T
V
W
T is function from V to W,
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Let V and W be vector spaces over the field F
and let T be a linear transformation from V
into W.

• If V is finite-dimensional, the rank of T is the
  dimension of the range of T and the nullity of T
  is the dimension of the null space of T.

• The null space (kernal) of T is the set of all
  vectors x in V such that T(x) 0

V
W


T

ran T

0
ker T
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The Algebra of Linear Transformations

• Let T U ? V and S V ? W be linear
  transformations, with U, V, and
• W vector spaces.

The composition of S and V
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Representation of Linear Transformations by
Matrices ?????????
Let V be an n-dimensional vector space over the
field F. T is a linear transformation, and a1,
a2,,an are ordered bases for V. If
?A?Linear Transformation T?a1, a2,,an ????
??
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Inner Product ????
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y
z
b - a
a
a2
b
a3
?
a
y
a1
x
x
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Inner Product Space ??????
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Eigenvalues and Eigenvectors ????????
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T
1
1
2
3
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Diagonalization ???
A square matrix is said to be a diagonal matrix
if all of its entries are zero except those on
the main diagonal
A linear operator T on a finite-dimensional
vector space V is diagonalizable if there is a
basis vector for V each vector of which is an
eigenvector of T.
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Orthogonalization of Vector Sets ??????
Gram-Schmit orthogonalization procedure
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Quadratic Forms ????
A x y
Equivalent
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Canonical Form ????
?Diagonal matrix
If the eigenvalues and corresponding eigenvectors
of the real symmetric matrix A are known, a
matrix Q having this property can be easily
constructed
eigenvector
eigenvalue
Let a matrix Q be constructed in such a way that
the elements of the unit vectors e1, e2,.,en are
the elements of the successive columns of Q
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Ex Let T be the linear operator on R3 which is
represented in the standard ordered basis
by the matrix A
? Orthogonal matrix