Post on 05-Apr-2018
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Portfolio of Rebekah Schumacher
Spring 2011
Mathematics PortfolioRebekah SchumacherSpring 2011
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Statement about MathematicsMathematics is a challenge. Perhaps this is what draws me to it. Those that claim
that mathematics is difficult are absolutely correct, yet this never deterred myresolution to complete my math courses. Despite their intensity, I learned that I
loved the solidity of working through problems and having a concrete feeling
of satisfaction when I arrived at the correct answer. What perhaps intrigued me
most about mathematics was that regardless of its difficulty, there is a certain
simplicity and artistry present as well. It has been said that mathematics is the
language of science, and I believe that more now than ever before. Lookingpast the computational requirements, the intricacy of how math is intertwined
with almost every other field has always amazed me.
The opportunities for growth in mathematics are endless. There is no cap on
knowledge. How much I am able to comprehend or understand is solely my
decision. There is little subjectivity in mathematics for either I understand it or
I do not, and the ability to have this kind of control over how well I perform in
my work is satisfying. The certainty of mathematics is ultimately what has
drawn me to this field, and I am pleased to have a chosen a subject that
challenges my scope of knowledge.
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Portfolio Elements
Basic Content
Advanced Content
Mathematic Modeling and Applications
Proofs
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Basic ContentIntegration by u-substitution
Integration by substitution is aprocess by which one indefinite
integral is transformed into another;
however, the second integral is
simpler than the first, making thebasic computation easier. The first
step in u-substitution is to choose
your u which requires practice. In
this example, sin x is chosen. Then,
du is written as the derivative ofu asillustrated here by cos x. Then one
integrates the simpler indefinite
integral before substituting back to
eliminate u.
Substitution is useful for
it allows us to find the
antiderivative of a
function which is often
required when working
with the Fundamental
Theorem of Calculus.
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Basic ContentIntegration by trigonometric substitution
A special form of u-substitution that is helpfulwhen integrating is trigonometric substitution.
Often integrals that contain roots of quadratic
expressions are ideal candidates for
trigonometric substitution even if this is not
obvious at first glance. Consider the example tothe right. Carefully choosing ourx to be tan u
provides us with the trig identity to the right.
Substituting this identity into our integral
allows us to obtain the most simple integral.
Choosing the proper trig identity or substitutionis the most challenging aspect of these
integrals; however, if chosen correctly, they
can reduce integrals that contain roots of
quadratic expressions to powers of trig
functions.
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Basic ContentAbsolute extrema of a function on a closed interval
Finding the absolute extrema of a
function, i.e. min and max, is a
component of a highly applicable
realm of mathematics known as
optimization. Using the example onthe right, one must first take the
derivative of the function and then
find the critical points of the function
by setting the derivative equal to 0.
Evaluating the function at the
endpoints yields the absolute min and
max. Furthermore, taking the second
derivative of the function and
evaluating it at the critical points will
yield the local min and local max.
Being able to find maximum andminimum values of functions in the
real world is a highly valuable
application of calculus.
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Basic ContentDetermining convergence/divergence of series by the Ratio Test
Determining the convergence or divergenceof a series is possible by using many
different tests, one of which is the Ratio
Test. This test states the following: Suppose
that for all a > 0 for all k and that
Then for all L < 1, the series converges;
for all L >1, the series diverges; for
L=1, the test is inconclusive whichindicates that either convergence or
divergence is possible. The example
given above demonstrates that the given
series diverges after algebraic
manipulation converts it into a form to
which the Ratio Test can be applied.
The Ratio Test is often more simple to
use in place of the Comparison Test;
however, in the event that L=1, the
Comparison Test should be used
instead. The Ratio Test is also
particularly beneficial for power series.
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Basic ContentEquivalence Relation
An equivalence relation as we have learned in discrete mathematics is a
relation who has the qualities of being reflexive, symmetric and transitive.
The reflexive property is defined by the following: For x X, R is reflexive if
x R x. For the symmetric property: R is symmetric if for every x R y, then
y R x. For the transitive property: R is transitive if for every
x R y and y R z, then x R z.
To demonstrate, consider the following relation:
R = {(a,a), (b,c), (c, b), (d,d)}
First, we check for the reflexive property. It is clear to see that this relation
does to hold the property since there exists no (b,b) or (c,c).Next, we check the symmetric property. The relation is symmetric since for
(b,c), (c,b) does exist.
Lastly, we check the transitive property. It is also evident that this relation is
not transitive since for (b,c) and (c,b), (b,b) does not exist.
Hence, the relation is not an equivalence relation since it does not fufill allthree properties required.
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Basic ContentProbability of Winning the Lottery
To understand the possibility of winning the lottery from a mathematical
perspective requires the use of probability. Let us consider the following
example: Suppose that in a local lottery game, a prospect is required to choose
three numbers between 0 and 9. In a particular type of box play win, three
distinct numbers chosen by the prospect must match those drawn by the lottery
representative with repetitions permitted. If the prospect were to choose three
distinct numbers, then the probability of winning the lottery is as follows:
Here, 10
3
represents all the total possibilities of matches between theprospect and the lottery representative, and 3! represents the total number of
combinations that the prospect is able to choose. Dividing the latter from
the former obtains the prospects prospects, or 0.006 chance of winning the
lottery.
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Basic ContentThe inverse of a 2 x 2 matrix
The inverse of a matrix is often useful in solving matrixequations. Given a 2 x 2 matrix, one way of solving to obtain the
inverse is through Gauss-Jordan elimination. Examining the
following matrix will show this process:
The first step is to write the matrix A directly
next to the identity matrix, then using the steps
of Gauss-Jordan elimination, eventually obtainthe identity matrix on the opposite side. To
ensure that the resulting inverse matrix is
correct, one need only multiply AA-1 and
check that the identity matrix is the result of
the multiplication.
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Basic ContentCross Product
Within Linear Algebra, a type of vector multiplication that is used to produce a
vector as a product in 3-space is the cross product. The definition of this product
is that given two vectors u=(u1, u2, u3 ) and v = (v1, v2, v3 ) in 3-space, the cross
product u x v is defined by
or in determinant notation by
Consider the simple example
of u x v where u = (1, 2, -2)
and v = (3, 0, 1). Using the
cross product, the following
is obtained:
One final note regarding the cross product is that while both the cross product
and the dot product are two types of vector multiplication used in 3-space, the
difference between the two is that the cross product is a vector while the dot
product is a scalar.
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Advanced ContentIntegrating Factors for Linear Differential Equations
Rather than simply guessing inorder to solve first-order
nonhomogenous linear differential
equations, another method that is
used is that of integrating factors
which is a more analytical
approach. The idea is that anintegrating factor often known as
(t) can cause the differential
equation to take on the shape of the
derivative of a product of two
functions. If an integrating factor
that satisfies the equation can be
found, then a new differential
equation is created. Both sides of
this new equation can be integrated
with respect to t, allowing the
computation of the general solution
y(t).
Of course, it should be noted that this only will
work if the integrating factor can be found and if
the integration is actually possible. However, a
formula has been found that allows easy calculation
of the integrating factor by using
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Advanced ContentNormal Subgroups
The definition behind a normalsubgroup is that given a subgroup H
of G, H is considered a normal
subgroup of G iff xhx-1 H for every
h H and every x G. Suppose wewere asked to check whether the
subgroup is a normal
subgroup of the multiplicative group
G of invertible matrices in M2(R).
To do so, we show the following:
Since aH=Ha, then for alla G aHa-1 = H x-
1Hx = H.
While this is a simple
example, normal
subgroups are important
for two main reasons. The
first is that they are
precisely the kernels ofhomomorphisms, and the
second is because they
can be used to create
quotient groups from agiven group.
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Advanced ContentNeighborhoods
In the realm of Real Analysis, a neighborhood is based on theprinciple of the closeness of two points such as x and y. Since
this closeness can be measured by the absolute value of their
difference, a neighborhood of x can be defined as a set of the
form where x R and > 0. Here, issome positive measure of closeness and can be referred to as
the radius of the neighborhood. Neighborhoods are often used
when describing the concepts of open and closed sets; hence,
studying these sets is an aspect of a field known as point set
topology. In essence, neighborhoods are open sets. With this inmind, neighborhoods are infinite as other neighborhoods have the
ability to be inside each other. Real analysis provides a context in
which we think of space and size quite differently, and
neighborhoods are a perfect example of a different type of space.
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Mathematical Modeling and ApplicationsUndamped Forcing and Resonance Beating
Harmonic oscillators are real world mechanical systems that when displaced from
equilibrium will experience some type of restoring force that is equal to the displacement
(this is explained more clearly by Hookes Law). The equation that will be given below
represents a harmonic oscillator that is forced and undamped. Although it may not seem
intuitive at first why one would study undamped harmonic oscillators since all systems of
this type have some damping, it actually does provide insight into oscillators or other
systems where the damping is small and provides a close approximation.
There are many cases of undamped equations; however, one that is of particular interest is
sinusoidal forcing. In this kind of forcing, something known as resonance occurs when the
frequency of the forcing function approaches the natural frequency of the equation. The
type of resonance that will be discussed below is a phenomenon known as beating. Thisoccurs when the frequency of both the natural response and the forced response are
approximately the same.
Thus, consider the following undamped equation:
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Mathematical Modeling and Applications
Undamped Forcing and Resonance - Beating
The first step is to find the generalsolution of the differential
equation. This is shown in steps
(1), (2) and (3). To find it, we use
the Method of Undetermined
Coefficients. Part of the solutionwill be real and part imaginary;
however, for this example, the
focus will be on the real part of the
solution with the forcing factor
cos(wt). Once we have found the
general solution, the next step is to
determine the period and
frequency of the beats . This is
done using complex exponentials
as is shown in the next slide.
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Mathematical Modeling and Applications(continued)
Using and as exponentials, we then
use Eulers formula to think of the realpart of our complex function. Finally,
after manipulating exponents , we
calculate the desired real solution and
obtain the frequency and period of the
beats. The small frequency and the long
frequency are both present in the
beating and are best represented by the
illustration below. Note that the values
below are not the same values from the
equation we just demonstrated. This
picture is simply to demonstrate what
beating looks like. In a real world context, this beating is what youmight hear should you be listening to a piano or to
a guitar that may be out of tune. When an
instrument (particularly guitar) is being tuned, you
might recognize it as the wah-oo-wah-oo noise
that often accompanies it. This is just one example
of a form of resonance that can take place in aundamped forced physical system.
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Mathematical Modeling and ApplicationsRSA Encryption
RSA encryption (named for its creators Ron Rivest, Adi Shamir and LeornardAdleman) has been one of the most successful public key cryptosystems invented
to this day. The RSA alogrithm is based on the difficulty of being able to factor
large prime numbers. Its uses in the real world are plentiful as it is often used to
protect data such as computers passwords, digital signatures and e-commerce. The
algorithm works as follows:
- The first step is to compute the public key needed for the algorithm by choosing
two distinct prime numbers p and q.
- Next, we compute the product of the numbers: n = pq. This value m is known as
the public key because it is made accessible to the public, and it can also be
referred to as the modulus of p and q.
- The next step is to compute what is known as (n) = (p-1)(q-1).- Then we compute an integer e where it must be greater than 1 but less than (n)
such that the greatest common divisor of e and (n) is 1. This integer e is another
public key component of the cryptosystem.
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Mathematical Modeling and Applications(Continued)
Now, the next step in the RSA algorithm is to use the Euclidean algorithm to find the private keywhich we will call d. This integer is computed by the following formula: e d mod (n) = 1. Next, to
encrypt the message, we use c = me(mod n) where the receiver already knows the public keys e and n.
When the encrypted message is send back to the receiver by the sender, the receiver can only decrypt
the message by using m = cd(mod n); however, the receiver must know the private key d or trying to
decrypt the message it futile. The following is a simple example of the algorithm:
Choose p = 5 and q = 11; hence, pq = 55.Finding (n) = (p-1)(q-1) = (4)(10) = 40. Since e must be relatively prime to (n) in order for it to be
the key, then in this particular case, any key e that is not divisible by 2 or 5 will have a matching key d
since the least common multiple of (p-1)(q-1) is 20. Hence, let e = 7.
To compute d, we use (ed-1) mod (n) = 0 which produces the key d = 3.
Choosing m = b = 2 as our plaintext message (or message to be encrypted), we use the formula c =
me(mod n) which becomes c = 27(mod 55) = 18.
To decrypt the ciphertext message (or message to be decrypted), we use the formula m = cd(mod
n) which becomes m = 183(mod 55) = 2.
This example illustrates the algorithm with very simple prime numbers; however, it is recommended
today that the prime numbers be at least 2048 bits in length to ensure security. Factorizing large prime
numbers is difficult and can take years. To date, the longest bit prime number that has been factorized
was 768 bits long, and the factorization occurred in 2010. Due to its difficulty, the RSA algorithm hasproven to be a secure way to protect online information.
P f
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ProofsBy contradiction
Proofs by contradiction sometimes allow us to draw conclusions about our statements
in a much easier than way than by direct proofs. To illustrate a proof by contradiction,
consider the following:
For x,y R, prove that if x is a rational number (x Q) and y is an irrational number
(y Q), then (x+y) is also irrational ((x+y) Q).
Proof: Suppose that x Q and y Q. Also suppose that (x+y) Q.By the definition of a rational number, x Q means where a,b Z
and b = 0. Also, (x+y) Q means where m,n Z and n = 0. Then,
Since Z is closed under substraction, mb-an , nb Z and nb = 0.Hence, y Q. However, our assumption was that y Q so we have a contradiction
(=>
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ProofsInduction
Mathematical induction allows us to make statements about the natural numbers
without having to verify the statement for each individual number which would proveimpossible. Suppose that we want to prove the following statement regarding the
natural numbers:
To begin the proof, we establish are base case to prove that P(1) is true:
Next is the induction step which will prove that the P(k) is true:
Thus, by the principle of mathematical induction, P(n) is true for all n.
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ProofsThe Archimedean Property
The Archimedean Property is a consequence of the Completeness Axiom (recall that the
Completeness Axiom states that a non-empty subset of R that has an upper bound alsohas a supremum), and it claims that the natural numbers are unbounded above in R. Its
proof depends on the completeness axiom and a contradiction. Thus to prove that N is
not bounded above in R, we begin as follows:
Suppose, to the contrary, that there is an upper bound on N.
Hence, the Archimedean Property is true based on the Completeness Axiom and a proof
by contradiction.
P f
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ProofsSequence Convergence
One way to prove that a sequence converges is to use the definition of sequence
convergence. This definition states that for each > 0, there exists a real number N suchthat for all n N, n > N implies that |sn-s| < . If the sequence converges to s, then s is
the limit of sequence; if it does not, then the sequence diverges. This type of proof is
popularly known as an -N proof. To demonstrate, prove that the sequence
converges to 3.
Proof:
Although a short proof, using the definition of sequence convergence provides a simple
way to show that a sequence does indeed converge.