portfolio of rebekah schumacher

8/2/2019 Portfolio of Rebekah Schumacher

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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


4/23

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.


5/23

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.


7/23

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.


8/23

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.


9/23

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.


10/23

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.


11/23

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.


12/23

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


13/23

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.


14/23

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.


15/23

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:


16/23

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.


18/23

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.


19/23

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


20/23

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

(=>


21/23

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.

P f


22/23

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.

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