Graph Analysis Discussion
Discussion The graphs of the data presented that the percent coverage increased at a non–constant
rate until reaching around optimal temperature (38.5°C) and decreased afterwards. The graphs also
showed that the number of colonies initially increased to 25, but then decreased by 19 and then
increased by 13. For both the number of colonies and percent coverage, there was no bacterial
growth at 12°C. Consequently, there are no error bars because the percent coverage and colonial
growth was always 0 at this temperature. Both trend lines are represented as polynomial curves. The
equation for the graph comparing the number of colonies to temperature (y=.0025x3 – .2046x2 +
4.3706x) is an odd degree polynomial, meaning that the end behavior of ... Show more content on
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Both tests show that when temperature changes from 12°C to 22°C, more Bacillus subtilis will
grow. Between 12°C and 38.5°C, the growth of Bacillus subtilis increased, but the number of
colonies decreased. Because the t–Stat value of percent coverage (13.95) is greater than the critical
one–tail value (1.69), the null hypothesis should be rejected. Similarly, when performing a t–Test for
bacterial colonies, the t–Stat value of 4.47 was greater than the one–tail critical value of 1.69,
showing that at 38.5°C, there will be fewer colonies than at 12°C. However, as temperature
increased, colony size did as well. Therefore, as temperature increased, so did the amount of
Bacillus subtilis present. Based on the t–tests between 12°C and 49.5°C, it is determined that the
percent coverage of bacteria increases whereas the number of colonies decreases. In the t–Test of
the percentages of bacterial growth, the t–Stat value of 11.6 is larger than the critical value of 1.69,
and in the t–Test of the number of colonies of Bacillus subtilis, the t–stat value of 8.90 is greater
than the one–tail critical value of 1.69, presenting that the null hypothesis should be rejected in both
cases. Although the number of colonies decreases, the size of the colonies increased as temperature
did, further revealing that there is correlation between temperature change and
... Get more on HelpWriting.net ...

Math Hl Type 1 Shadows
Higher Level Mathematics Internal Assessment Type I Shadow Functions Contents Introduction:
Functions/Polynomials 3 Part A: Quadratic Polynomials 4 Part B: Cubic Polynomials 12
Introduction: In mathematics, function is defined as a relationship, or more of a correspondence
between the set of input values and the set of output values. Also, a rule is involved, or as it may be
referred to, a 'set of ordered pairs' that assigns a unique output for each of the input. The output
correspondence is usually defined as f and the output is x. The correspondence is denoted as f(x).
All functions are mainly defined by two factors, as was mentioned before, set of inputs – which are
called arguments; and outputs – which ... Show more content on Helpwriting.net ...
So it can be said that: –(x+2)2–9+18 = –Y1+2Ym = Y2 At this point, the real and imaginary
components of the complex zeros of Y1 can be found using the values for zeros of Y2. As it is
known, Y1 has complex zeros of a form of a±ib and Y2 has zeros of a±b. With this information
available, I will show you how to obtain value for complex zeros of Y1. Consider a function Y1,
with a general statement of: Y1 = (x–a)2+b2 As it has been found earlier, this function has zeros of
a±ib. It also has a shadow function Y2: Y2 = –(x–a)2+b2 With zeros of a±b. If we say that a = 2 and
b = –3, then this function has zeros of 2±(–3). In this form of zeros, we can say that 2 is the x value
of the vertex coordinates, lying on the axis of symmetry of Y2 on the x–axis, and ±3 are just the
distances between the mid point 2 to the points where Y2 intersects x–axis. It is clearly shown on
the graph below. As we know the zeros of Y2, it will be very easy to determine the complex zeros of
Y1. 2 will equal to the real part of the complex zeros of Y1, and subsequently, ±3 will equal to the
imaginary part of the complex roots. Although, as we know that y–axis is the imaginary axis, the
imaginary parts of the complex roots obtained, of the Y1, will be plot according to the y–axis. As
you may have observed, in order to get the complex zeros of

What Is Optimized Design
In most parameter tables, the data values or keys are stored discretely and unevenly [13]. However,
parameter values do not generally exist explicitly in data tables. Thus, searching is processed for the
stored data keys. This chapter explains the details of the optimized design, the algorithm framework
and the advantages. 6.1 Initial Step In this algorithm, the first step is differentiated from the
remaining steps. Let N be the initial n most significant bits of the input bits A. Let k be the length of
the generator polynomial K(x) used for generating the CRC value. The binary numbers of N and A
are given by: N=[a1a2.....an] and A=[a1a2....ak] (6.1) The binary number of N is ... Show more
content on Helpwriting.net ...
Each slice uses a separate table. The slices are represented by M_1^(t–1),M_2^(t–2).....M_t^(t–1)of
length m1,m2...mt. The tables may vary from the tables used in the initial step. This is because the
number of bits read in the initial step may vary from the bits read in the subsequent steps. The step T
uses g tables of sizes equal to 2t1,2t2...2tg. Each table contains the remainders obtained from the
long division of all possible values of the slice with the offset value. The offset is given by
Oi=∑_(j=i+1)^g▒t_j (6.5) The linear search algorithm is introduced at this stage. This accelerates
the search speed to obtain values from the look–up tables. The values obtained from the input
stream are considered as the indexes for the table look–up. The index value is stored in a register
and is matched with every value present in the table. The output value is obtained when a match
occurs during the search. The values returned from the lookup tables is given by Pi=Mt–1i.
2oi.modG (6.6) Where P==_(i=1)^tPi (6.7) MT=PT.2q QT

Algebraic And Transcendental Equation
Solution of Algebraic & Transcendental Equation
Introduction
When mathematically modeling real life situations we often come across equations in the form of
f(x)=0. These equations can either be in the form of algebraic or transcendental equations. In certain
cases, these equations may prove to be difficult to solve, as the equation may not have an exact
answer. A problem of great importance in science and engineering is that of determining the
roots/zeros of an equation of the form
f(x) =0
A polynomial equation(algebraic equation) f(x)= Pn(x) = a0xn + a1xn–1 +a2xn–2 + – – – – – – – –
– – – – – – – – – – – + an–1x + an
An equation that contains polynomials, exponential functions, trigonometric functions etc. ... Show
more content on Helpwriting.net ...
e1 = –2.17798
f(0).f(1)< 0
Therefore root lies between .31467 & 1
x0 = .31467 x1=1
f(x0)=.51987 f(x1)=–2.17798
x3= x0 – [( x1 – x0) / ( f(x1) – f(x0) ) . f(x0)
= .31467 – [ (1 – .31467)/(–2.17798 – .51987) ] (0.51987) = .44673
f(x3) = f(.44673) = .20356 (+ve.)
f(.44673). f(1) < 0

Therefore root lies between .44673 & 1
x0 = .44673 x1=1
f(x0)=.20356 f(x1)=–2.17798
x4= x0 – [( x1 – x0) / ( f(x1) – f(x0) ) . f(x0)
= .44673 + [ .55327/ 2.38154 ] (.20356)
= .49402
Repeating this process
x5 = .50995
x6 = .51520
x7 = .51692
x8 = .51748
x9 = .51767
Hence the root is .518 correct to three decimal places.
Newton – Raphson Method
This method is also called Newton's Method or Chord Method
Let x0 be the initial approximation to the root of f(x) = 0
Then P(x0,f0) is a point on the curve . Draw the tangent to the curve at P the point
Of intersection of tangent with the x–axis is taken as the next approximation to
The root.
This process is repeated until the required accuracy is

What Is An Interpolation Equation
1. INTRODUCTION OF LAGRANGE POLYNOMIAL ITERPOLATION
1. 1 Interpolation:
First of all, we will understand that what the interpolation is.
Interpolation is important concept in numerical analysis. Quite often functions may not be available
explicitly but only the values of the function at a set of points, called nodes, tabular points or pivotal
points. Then finding the value of the function at any non–tabular point, is called interpolation.
Definition:
Suppose that the function f (x) is known at (N+1) points (x0, f0), (x1, f1), . . . , (xN, fN) where the
pivotal points xi spread out over the interval [a,b] satisfy a = x0 < x1 < . . . < xN = b and fi = f(xi)
then finding the value of the function at ... Show more content on Helpwriting.net ...
. . + ________________________________________ f4 (x0 – x1) (x0 – x2)(x0 – x3)(x0 – x4) (x4
– x0)(x4 – x1)(x4 – x2)(x4 – x3)
(0.3 – 1)(0.3 – 3)(0.3 – 4)(0.3 – 7) (0.3 – 0)(0.3 – 3)(0.3 – 4)(0.3 – 7) =
________________________________________ 1+
________________________________________ 3 + (–1) (–3)(–4)(–7) 1 x (–2)(–3)(–6)
(0.3 – 0)(0.3 – 1)(0.3 – 4)(0.3 – 7) (0.3 – 0)(0.3 – 1)(0.3 – 3)(0.3 – 7)
________________________________________ 49 +
________________________________________ 129 +
3 x 2 x (–1)(–4) 4 x 3 x 1 (–3)
(0.3 – 0)(0.3 – 1)(0.3 – 3)(0.3 – 4)
________________________________________ 813
7 x 6 x 4 x 3
= 1.831
Figure 1: Example of Lagrange interpolation 4. APPLICATIONS OF LAGRANGE POLYNOMIAL
INTERPOLATION

 Lagrange polynomials basis are used in the Newton–Cotes method of numerical

The Egyptian And Babylonian Mathematicians
Abstract–Research compiled from video lectures and articles retrieved from the internet is the basis
for the findings in this article related to solving a cubic equation. The noteworthy mathematicians
and their contributions to the solution and their understanding of the cubic equation is included.
Also included is an example of a cubic equation solved using Descartes' Factor Theorem.
Index Terms–complex number, cubic equation, Descartes, Riehmen Sphere, Tartaglia
Introduction
Building on the successes of their ancient predecessors the mathematicians of the European
Renaissance searched for an algebraic solution to the cubic equation. The ancient Egyptian and
Babylonian mathematicians produced solutions for the linear and quadratic equations. By 628,
Brahmagupta, the Indian mathematician, developed the general quadratic formula for solving a
quadratic equation. In the eighth century, the great Persian mathematician, Al–Kharizmi, offered a
solution to the quadratic equation by completing the square. But solving the cubic equation or
finding the zeroes of the polynomials of degree three evaded the great mathematicians. Omar
Khayyan, the Islamic poet, astronomer, and mathematician attempted to find a general algebraic
solution to the cubic equation but was able to only offer a geometric solution for a specific cubic
equation. During the Renaissance, Tartaglia, Cardano, Viete, Fermat, and Descartes, made advances
in solving the cubic equation. Later, Newton and Riemann would

Applying The Following Method To Calculate The Optimal...
CE2.18 To increase the BER performance, I applied the following method to calculate the optimal
rotation angle (θ_opt) for the interleaving system applied to MQAM/MPSK schemes. The selection
of optimal rotation angle is based on finding the expression of average symbol error probability
(P_s) corresponding to θ given certain signal–to–noise ratio (SNR). I found out that the optimal
rotation angle (θ_opt) approximately around 8.6 degree will maximize the BER performance.
CE2.19 I noticed that the simulating process was a time–consuming process due to iterative
decoding. I observed that for the calculation of probability of the binary index for kth bit, large
computational load required. As a result, memory usage was increased due to ... Show more content
on Helpwriting.net ...
I designed the normal BCH encoding process by using three steps using the above figure. 1) I used a
generator polynomial according to the DVB–T2 standard table the error correcting capability of
DVB–T2 is equal to 12. 2) I divided the message sequence using the generator polynomial in a
Galois field with α=2 and the remainder resulted in the parity check sequence. 3) I appended the
parity check sequence to the message sequence to get the BCH encoded message C2.25 I calculated
fast encoding process using G(X) = [Ik ⋮P]. The size of G(x) is Nbch × Kbch whereas letter "I" was
the identity matrix with the size of Kbch × Kbch and P was the redundancy matrix , the size of the
matrix was Kbch × (Nbch − Kbch) and Nbch was the length of the binary BCH code. C2.26 I was
concerned about the implementation of the fast encoding process as it was required to build a huge
matrix, which would over exceed the range. I reshaped this huge matrix with the help of Matlab
tool. Based on the simulation results, I concluded that longer parity–check will provide better the
error detection as well as correction and can pre–calculate the parity check sequence. C2.27 I was
assigned to design the BCH decoder. I created a syndrome generation by substituting 2t roots of the
original message in the polynomial form of the received codeword. I used these

Nt1310 Unit 6 Powerpoint
A. Setup: The setup phase takes input a security parameter . It selects a bilinear group of prime
order p with b as generator, and bilinear map The universe attribute is . It selects for attribute n, ,
and a random exponent . The public key and master key is given by (1) (2)
Though is publicly known to all system parties, is kept secretly by trusted authority (TA).
B. Key Generation: The key generation phase takes set of attributes S as input and the secret key
equivalent to S is produced as output. Initially, it selects a random number from . Then, it calculates
the key as (3)
C. Encryption: The

The Security Problems Faced By Cloud Computing
Related work:
In order to overcome the security problems faced by cloud computing a new technology known as
Homomorphic encryption is being put to use. It is a type of encryption that allows all the
computational process to be made available to carry on cryptographic which is one on a plain text
using an algorithm. This is one of the most improved and effective technique used in today's
architecture that is related to communication systems. This encryption combined all the different
services that are needed together without letting out the data to any of these services. This
homomorphic encryption data can be transferred into cryptographic encryption by altering the
design. This change allows their encryption to be used in cloud computing ... Show more content on
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On the other hand any cryptographic encryption that's allows only limited arbitrary functions such
as multiple addition options but only one multiplicative option to perform on data is known as
partially homomorphic encryption. When compared with complete homomorphic encryption partial
encryption is much faster and more intact. Although this encryption is open to attacks and can be
attacked at its base but if this encryption is properly used then it can perform computations in a
secure environment. One of its important features is to secure data on cloud. It allows customers to
feel secure that their data has been protected on cloud but at times when the customer wants to edit
their data then the data needs to be decrypted by the cloud service provider. The entire security of
your data is lost and the cloud provider who is in procession of the secret key can alter your
personal data. The idea of homomorphic encryption started way back but breakthrough in recent
times with complete homomorphic encryption would break down the problems faced by security in
cloud computing and would wide spread cloud adaption.
New solution:
Homomorphic encryption are studied widely after they became very important and pat of many
cryptographic encryptions such as in voting machines. Homomorphic encryptions

Algebra 1b
Hello it is your student Shems Haman from algebra 1b. I thought this unit was very important for
the future of algebra. One subject I learned from this unit is exponents. The teacher assigned us a
video to watch and I liked how they put all the technique of exponents in just one video. They said
there was six tools / techies for exponents. The first tool/ technique is the power of one. For example
x^1 then that would equal x because this rule says if nothing is with the exponent 1 it will always
equal itself. Another tool/ technique is if you have a negative exponent then it is it's inter verse. For
example if you have y^–4 then it is 1/y^4. This only works when y isn't 0. Multiplying exponent is
another technique. This technique is when there ... Show more content on Helpwriting.net ...
This technique is instead of adding the exponent like multiplication you just subtract them. For
x^3/x^2 = x^1. This only works if the variables are the same. The second to last technique is the 0
power. The 0 power says that if the exponent is 0 then it will always equal 1. Finally, the last
technique is if you have (x^3)^2 you multiple the exponents so it would equal x^6. Another subject I
leaned in this unit is earthquakes. What I learned about earthquakes and what it has to do with math
is there is a machine called ritcher scale and this calculates the power of the earthquake. This is
related to math because ritcher scale is logarithmic which means the magnitudes of earthquakes is
ten times stronger. For example if the magtuide is 0.1 then I would have to multiple by 10 so it
would be 1. Finally, the last subject I learned in this unit is polynomial. One thing I learned is
adding polynomial. How you add polynomial is for example if I had (3b+6) (5b+5), first you would
do 3b+5b = 8b+5+6 = 8b+11. You would do the same following steps for subtraction. Another
subject I learned is multiplying polynomial. How you multiple polynomial is for example (y+6)+
(y+6), first y*y= y^2 +6*6 =

Using Microsoft Excel And Microsoft Word
A quartic equation is a fourth–order polynomial equation of the form. Shortly after the discovery of
a method to solve the cubic equation, Lodovico Ferrari (1522–1565), a student of Cardano, found a
way to solve the quartic equation. His solution is a testimony to both the power and the limitations
of elementary algebra. The objective of this project is to analyze a polynomial of degree four via
various attributes using Microsoft Excel and Microsoft Word. I was assigned a nuumber in class that
resembles a fucntion provided on the list.
An end behavior on a graph basically talks about the tailends of the graph. You determine the end
behavior by looking at which way the tailends are pointing. The end behavior is the behavior of the
graph ... Show more content on Helpwriting.net ...
Looking at my graph i can come to a conclusion that the left side of my graph is going upward so
F(x) is approching positive infinity. The left side isnt pointing upward or downward so i had to look
at my equation to determine what f(x) is. Since the leading coefficient is negative the right side of
the graph is going down , so f(x) approches negative infinity.
Local extrema is basically all local maximums and minimums on a functions graph. Local extrema
occurs at criticla points on the graph where the derivative is zero or undefined. To find the exact
number of a local extrema using your polynomial , first find the first derivative of f using the power
rule. Then you will set the derivative to zero and solve for x. The values you get are the critical
points which is also your local extrema. To find the local extrema in your calculator all you have to
do is enter the equation into Y=. Then hit GRAPH and look for the max value first. To find the max
value just go to calculate and choose maximum. Then move your cursor to the left when they ask
you for left bound and hit enter and it will tell you that you locked the position and just repeat the
bounds for the right side too. When it asks for guess just hit enter and the coordinates of he max
value will appear. I had atleast 4 local extrema.
A zero of a function is an input value that produces an output of zero and can also be referred to as,
a root. An examples of this would be f(x)=

Track On The Millennium Force Roller Coaster
The purpose of this project was to find a function that models the track on the Millennium Force
roller coaster. During this project we created two functions to model this roller coaster, one of which
was a cubic function and the other was a quartic polynomial. Overall, the quartic polynomial
modeled the track of the roller coaster the best. It modeled the track the best for two main reasons.
The first being that the end behaviors of the quartic polynomial compared to those of the cubic
function. In the cubic polynomial the end behaviors are opposite of eachother. That means that as
the function approaches negative infinity it goes forever in the negative direction. But when the
cubic function approaches positive infinity the functions goes

Albert Reid Essay
Albert T. Reid was born in Hampton, Virginia to William Thaddeus Reid and Mae Elaine Beamon
Reid, on November 13, 1927 and passed away on February 26, 1985. Albert is now known as a
world–renowned master mathematician. Albert Reid was married to his beautiful wife, Rodab
Phiroze Bharucha in 1954. Reid also decide to adopt his wife's surname, Bharucha; giving him the
name, Albert T. Bharucha–Reid. He and his wife had two sons, Kurush Feroze Bharucha–Reid and
Rustam William Bharucha–Reid. Albert attended Iowa State University and obtained his Bachelor's
degree in Mathematics and in Biology in 1949 at the age of 19. Although Reid never completed a
graduate degree in his chosen field, he was still very successful in his career. Bharucha–Reid's area
of expertise was probabilistic analysis and its application and by 1956 he was employed in a
teaching position at the University of Oregon and in 1961 was associate professor of mathematics at
Wayne State University where he headed the Center for Research in Probability.
Albert found work as a research assistant and statistician at the University of Chicago, Columbia
University, and the University ... Show more content on Helpwriting.net ...
After rising to Dean and Associate Provost of Graduate Study at Wayne State, Bharucha–Reid
headed south in 1981 to become Professor of Mathematics at the Georgia Institute of Technology.
Two years later, he became a distinguished mathematics professor at Atlanta University. Albert was
also an advisor to at least 13 Ph.D. students. In 1984 he was awarded an honorary science degree at
Syracuse University. Albert Bharucha–Reid continued as editor of the Journal of Integral Equations.
He worked to advance the opportunities and recognition of minorities and women in the field of

The Effect Of Temperature On A Squash Ball
When a ball bounces, the kinetic energy is transformed into elastic potential energy. However, the
transfer of energy is not exactly perfect, as some energy is lost through heat and sound. The
coefficient of restitution is a formula that takes the square root of the ratio of bounce height to drop
height. The result ranges from 0 to 1, where 1 equals a perfect elastic collision.
In this experiment, the effect of temperature on a squash ball was investigated. Various types of
squash balls were subject to different temperatures then dropped from a 2 metre height. A slow–
motion capture camera was used to record bounce heights, then the results were carefully analysed
and recorded into the data table. The results showed that as temperature ... Show more content on
Helpwriting.net ...
It takes the square root of the ratio of bounce height to drop height. The result ranges from 0 to 1,
where 1 indicates a perfect elastic collision.
The coefficient of the restitution is given by the formula: e=√(h_after/h_before ) where h is the
height of the ball
This formula can be used to calculate the coefficient of restitution for all types of balls.
Since it is understood that a warmer ball will bounce higher than a cooler one, then it can be
concluded that temperature affects the coefficient of restitution. This is because the gas molecules
inside the ball expand as temperature increases, causing an increase in the energy of the molecules
bouncing faster inside the ball (Sheehan, 2015). In other words, as temperature increases, so too will
air pressure. Temperature also influences the elasticity of the ball. In physics, elasticity (from the
Greek word, "ductible") is the ability of a body to resist a distorting influence or deforming force
and to return to its original size and shape when that influence or force is removed (Landau,
Lipshitz, 1970). The coefficient of restitution is a helpful formula that measures the elasticity of a
substance: the less energy lost to heat and sound, the higher the coefficient of restitution and the
more elastic the substance.
Rubber elasticity describes the mechanical behavior of many polymers. The polymers of a squash
ball are stretched upon impact for a short

Who Is Elbert Frank Cox?
Elbert Frank Cox (December 5, 1895 – November 28, 1969) was an American mathematician who
became the first black person in the world to receive a Ph.D. in mathematics. He spent most of his
life as a professor at Howard University in Washington, D.C., where he was known as an excellent
teacher. During his life, he overcame various difficulties which arose because of racism. In his
honor, the National Association of Mathematicians established the Cox–Talbot Address, which is
annually delivered at the NAM's national meetings. The Elbert F. Cox Scholarship Fund, which is
used to help black students pursue studies, is named in his honor as well. In 1917 after graduating,
Cox joined the U.S Army in World War I. After he discharged from the Army, he began his career as
a high school math tutor. Besides mathematics, Cox took courses in German, English, Latin, history,
hygiene, chemistry, education, philosophy and physics. Cox's brother Avalon was at Indiana
University as well; there were three other black students in his class. He received his bachelor's
degree in 1917, at a time when the transcript of every black student had the word "COLORED"
printed across it. After serving in the US Army in France during World War I, he returned to pursue
a career in teaching, as an instructor of mathematics at a high school in Henderson, Kentucky. In
December 1921 ... Show more content on Helpwriting.net ...
He expanded on the work Niels Nörlund had done on Euler polynomials as a solution to a particular
difference equation. in particular, Cox introduced generalized Euler polynomials and the generalized
Boole summation formula as an expansion on the Boole summation formula. He also studied a
number of specialized polynomials as solutions for certain differential equations. In his other paper,
published in 1947, he mathematically compared three systems of grading. The site points out that
one common use of polynomials in everyday life is figuring out how much gas can be put in a

Computing Of Data And Homomorphic Encryption Essay
Introduction Cloud Services have become more popular as they provide a lot of advantages like
high speed processing ,Flexibility and Disaster recovery.The problem is Security of data and how to
ensure that data being processed at the cloud is secure The motivation behind choosing this topic is
the many advantages of Computing of encrypted data and homomorphic encryption (HE) like
Delegation which is when a client can delegate the process of data to the powerful third party
(server) while still maintaining data privacy. To this end, the client could send the server an
encryption of the data, created employing an HE scheme. The server is able to run processes over
the encrypted data, and return an output to the client; the client needs only to decrypt to receive the
processed answer. The server here can actually be a collection of computing devices (cloud). In
addition to delegation, remote file storage can be more than ever secure with Homomorphic
encryption(HE), In a motivating example, consider a user that wants to run a keyword search on its
entire set of encrypted data. Without HE, since the server can't tell which documents contain the
keyword, it would be forced to send the entire set of encrypted data back to the user, who could
decrypt it and look for the keyword. With HE, however, the server can simply run the keyword
search algorithm with the encrypted keyword and the set of encrypted data, and send an encrypted
list of documents containing the keyword back to the

Psy 315 Week 4 Mat Analysis
A1.
The rubric aligns with the NCTM Assessment Standards in the following ways:
a. Mathematics: The rubric used with the assessment checks for students' understanding and work
process through all problems presented on the quiz: do they understand the concept? Are they able
to follow the process correctly? The rubric focuses on John's thought and reasoning process.
b. Learning: By applying the rubric to the assessment, I am able to see Johns' understanding or lack
thereof. His work allows me to see where he is in terms of progression of the concept while also
allowing me to see the need for adjustments to instruction. Because I am able to see his work, I can
go ahead and make decisions regarding and instruction adjustments that may need to ... Show more
The student's work on fractional exponents, radicals, and factoring is missing which leads to the
wrong answer.
Process 4 The student shows comprehensive understanding of laws of exponents and their
application. The work demonstrates correct setup and application of different rules that pertain to
each problem. Manipulation of different components for each problem leads to correct answer.
3 The student shows reasonable understanding of laws of exponents and their application. The work
demonstrates minor errors in setup and application of different rules that pertain to each problem.
Minor errors in manipulation of different components for each problem leads to wrong answer.
2 The student shows limited understanding of laws of exponents and their application. The work
demonstrates many errors in setup and application of different rules that pertain to each problem.
Many errors in manipulation of different components for each problem leads to wrong answer.
1 The student shows a lack of understanding of laws of exponents and their application. The work
on setup and application of different rules that pertain to each problem is missing. Student is unable
to manipulate different components for each problem which leads to wrong

Uses And Accuracy Of Newton 's Method
Newton's Method Introduction Sir Isaac Newton is famous for many discoveries in both math and
science. From gravity to calculus, Newton made many fundamental breakthroughs that have shaped
thought for centuries and are still in use today. For this reason, Newton has always been one of the
most interesting characters in history for me and thus is why I found such great interest in his
theories and being able to explore them further. However, though he had countless monumental
breakthroughs, he also had other theories that are not as well known, which interested me even more
as they are not discussed as much in school. One such theory is Newton's method for approximating
the zeroes of a function, which is simply known as Newton's Method. This method is a unique
approach for approximating zeroes due to its use of several iterations of a formula to slowly grow
closer and closer to the zero. The aim of this paper is to investigate the use and accuracy of
Newton's Method to approximate the zeroes of a function. This investigation aims to explore the
history of the method and possible influences to Newton's discovery, the basic mechanics of the
method such as how and when it works, and finally Newton's method will be compared to the most
conventional means of determining the zeroes of function which are the algebraic formulas like the
quadratic formula. History Although it was Newton who would eventually be credited with the
discovery of a method for finding the roots of a

The Importance Of Computers In Communication Codes
As motivated above, it is often necessary to be able to reliably send data over noisy or otherwise
unreliable communication channels. We can use error correction codes to help mitigate the
introduction of errors into our data. To do this, we can divide our data into smaller pieces called
messages and map these messages to codewords through a process called encoding. This process
will provide necessary redundancy to correct a certain number of introduced errors during the
decoding process. An (n, k) block code C is an injective map E : Q k 7→ Q n where Q is a set of
symbols called our alphabet. The size of our alphabet, |Q| is denoted as q. In many practical
applications q = 2 where Q = {0, 1}. E(m) = c maps our message m of length k to our ... Show more
. . , k, a contradiction. Thus, h(x) = 0 and f(x) = g(x). 3 Thus, the added n − k points provide
redundancy in case an error occurs during transmission of our message. We'll write our received
word as c 0 = (y0, y1, . . . , yn−1). Suppose that an error does occur. Theoretically, we could decode
the message by taking all subsets of size k of (y0, y1, . . . , yn−1) and interpolate each of the k
numbers in each subset to determine a polynomial of degree k − 1. We could then find which
polynomial occurs most often. The coefficients of that polynomial would then be taken as the
original message [9]. Obviously we cannot perform this decoding procedure due to computational
constraints. We would need to find n k polynomials which is infeasible for even slightly large
values of n and k. For this reason, the original formulation of the codewords was changed. Let our
alphabet Q be a finite field Fpm for some prime p generated by f(x) with primitive element α. We
construct for our code a generator polynomial g(x) whose roots are α, α2 , . . . , αt where t = n − k.
In other words g(x) = Y t i=1 (x − α i ). Again, let p(x) = Pk−1 i=0 mix i . To achieve a systematic
Reed–Solomon encoding (non systematic encodings exist, but will not be discussed here), our
codeword is defined as c(x) = x t p(x) − x t p(x)modg(x) (2) Notice that the first term in c(x) yields a
(n − k) + k − 1 = n − 1 degree polynomial where the lowest n − k degree terms have coefficient 0.
The

Summary: Unit 2 Of College Algebra
After applying more of my critical thinking in solving some exercises while reading has been of a
great effect to my learning process in unit 2 of College Algebra. According to Stitz, C. Zeager, J.
(2011), During the second week of term 4, I could read and cover many topics but the text I found
difficult to read are firstly, Graphs of Polynomials that states where a0, a1, . . . , an are real numbers
and n _ 1 is a natural number. According to 3.2 definition where we can now think of linear
functions as degree 1 (or first degree') polynomial functions and quadratic functions as degree 2 (or
second degree') polynomial functions. The end behavior of a function is a way to describe what is
happening to the function values (the y–values) as the x–values approach the `ends' of the x–axis.9
That is, what happens to y as x becomes small without bound and, on the flip side, as x becomes
large without bound. While reading to understand this concept, I took up and practice this
convention as stated in the text at this point and considered when plotting it graphically, it looks a
bit complex to my understanding. All ... Show more content on Helpwriting.net ...
The zeros of p are the solutions to x2 +1 = 0, or x2 = –1. The imaginary unit i satisfies the two
following properties. Establishing that i does act as a square root2 of –1, and property 2
establishes what we mean by the `principal square root' of a negative real number. After reading the
textbook for hours, I had no choice than to repeat the reading the passages twice because at the
initial stage, I barely could understand every mathematical functions, calculations, and graph of
polynomial. I was able to read the text book twice in other to understand most part of the Graphs of
polynomial, The Factor Theorem and The Remainder Theorem, and the Complex Zeros and the
Fundamental Theorem of

Who Is Amalie Emmy Noether?
Amalie Emmy Noether was born on March 23, 1882 to a Jewish family in Erlangen, Bavaria,
Germany. Emmy had three brothers Fritz Noether, Alfred Noether, Gustav Robert Noether. Her
father Max Noether was a mathematician professor. Emmy Noether spent an average childhood
learning the arts that were expected of upper middle class girls.
Noether graduated from Höhere Töchter Schule in Erlangen. In 1900 she was certified to teach
English and French. But rather than teaching, she pursued a university education in mathematics,
which was then considered as a challenging path for a woman. She took Mathematics classes for
two years from the University of Erlangen after obtaining permission from the German professors.
After passing the matriculation exam ... Show more content on Helpwriting.net ...
She developed the abstract and conceptual approach to algebra, which resulted in several principles
unifying topology, logic, geometry, algebra and linear algebra. Her works were a breakthrough in
abstract algebra. Her study based on chain conditions on the ideals of commutative rings were
honored by many mathematicians all over the world. Her paper 'Idealtheorie in Ringbereichen' or
'Theory of Ideals in Ring Domains', published 1921, became the foundation for commutative ring
theory. The 'Noetherian rings' and 'Noetherian ideals' formed part of her mathematical contributions.
Her insights and ideas in topology had a great impact in the field of Mathematics.
The third epoch began from 1927–1935, where non–commutative algebras, representation theory,
hyper–complex numbers and linear transformations became the primary focus of her study. Noether
was awarded the Ackermann–Teubner Memorial Prize in Mathematics in 1932.
Noether had undergone surgery to remove a uterine tumor, but she died from a post–operative
infection in April 1935. She was fondly loved and respected by her students. The University of
Erlangen honored her after World War II ended. A co–ed gymnasium, dedicated to Mathematics was
named after her in

Analysis Of Two Party Key Agreement Protocol
Two Party Key Agreement Protocol for MANETS
This model does not have a dealer, and set of participants, but has only source and destination who
wish to share a secret key between them because the aim of this model is to share a secret key
between the two end parties to communication. Source plays the role of dealer and destination plays
the role of set of participants. Source chooses the secret key to be shared with the destination,
partitions the secret key into 'n' shares using Shamir's secret sharing scheme [24], where 'n' is the
number of disjoint paths exists between source and destination. Source calculates the 't'
commitments to the coefficients of 't–1'degree polynomial and transmits the shares and
commitments to the destination ... Show more content on Helpwriting.net ...
Let A and B wants to share a secret key K between them.
4.2.1. Modules
This module can be described as follows: Distribution Phase Verification Phase Reconstruction
Phase Key agreement Phase
Distribution phase In Distribution phase A chooses the secret key which is created from randomly
chosen values taken from 〖 Z〗_q^*. Both AB calculates shares and commitments by
themselves, they sends this calculated ones to the others vice versa in multiple paths as each path
holds one through it.
(1) A chooses S1 = [u]P where u is chosen at random from Z_q^*. Similarly, B chooses S2 = [v]P
where v is chosen at random from Z_q^*.
(2) A selects a polynomial p1(x) = a_0 + a_1x + ... + a_(t–1) x^(t–1) where a_0 = u and a_1, a_2...
a_(t–1) are randomly selected from Z_q^*.
B selects a polynomial p2(x) = b_0 + bx + ... + bx^(t–1) where b_0 = v and b, b_2... b_(t–1) are
randomly selected from Z_q^*.
(3) A calculates 〖CA〗_i = f 〖(P,P)〗^(a_i ) for i = 0 to t–1 as the commitments of S_1 and P_1
(x). These values are transmitted to B via multiple paths so that they certainly reach B.
(4) B calculates 〖CB〗_i =f〖(P,P)〗^bi for i = 0 to t–1 as the commitments of S_2 and P_2(x).
These values are transmitted to A via multiple paths so that they certainly reach A.
(5) A calculates the shares U_i = [p_1 (i)]P mod q for i = 1 to n and then sends the shares to B via 'n'
disjoint paths; one share thru each path.
B calculates the shares V_i = [p_2

Distributed Lag Model For Money Supply And Price...
CHAPTER EIGHT DISTRIBUTED LAG MODEL FOR MONEY SUPPLY AND PRICE
RELATIONSHIP 8.1 Distributed Lag Model The economic variable Y is affected by not only the
value of X at the same time t but also by its lagged values plus some disturbance term i.e.X_t,X_(t–
1),X_(t–2).....,X_(t–k),ε_t.this can be written in the functional form as: 〖Y_t=f(X〗_t,X_(t–
1),X_(t–2).....,X_(t–k),ε_t) In linear form, Y_t=α+β_0 X_t+β_1 X_(t–1)+β_2 X_(t–2)+⋯+β_j X_(t–
k)+ε_t (8.1) Where, β_0 is known as the short run multiplier, or impact multiplier because it gives
the change in the mean value of Y_t following a unit change of X_tin the same time period. If the
change of X_t is maintained at the same level thereafter, then, (β_0+β_1) gives the change in the
mean value of Y_t in the next period, (β_0 + β_1+β_2) in the following period, and so on. These
partial sums are called interim or intermediate multiplier. Finally, after k periods, that is =β,
therefore ∑▒β_i is called the long run multiplier or total multiplier, or distributed–lag multiplier. If
we define the standardized β_i^* = β_i/(∑▒β_i ) then it gives the proportion of the long run, or
total, impact felt by a certain period of time. In order for the distributed lag model to make sense,
the lag coefficients must tend to zero as k. This is not to say that 2 is smaller than 1; it only
means that the impact of X_(t–k)on Y_t must eventually become small as k gets large. The
distributed lag plays

Wireless Sensor Networks ( Wsns )
Abstract–Key management is one of the most important issues of any secure communication with
the increasing demand for the security in wireless sensor networks (WSNs) it is important to
introduce the secure and reliable key management in the WSNs.Data confidentiality and authenticity
are critical in WSNs. Key management objective is to secure and keep up secure connections
between sensor nodes at network formation and running stages.
In this paper we proposed various key management schemes, necessity for key management and
security requirements for WSNs and made a detailed study to categorize accessible key management
strategies and analyze the conceivable network security.
1. INTRODUCTION
At present, Wireless Sensor Network has becoming a hot technological topic with the development
of computer science and wireless communication technology, wireless sensor network (WSNs) is a
system shaped by a substantial number of sensor nodes, each one furnished with sensors to
recognize physical phenomena, for example, heat, light, movement, or sound. Utilizing diverse
sensors, WSNs can be implemented to backing numerous applications including security, diversion,
mechanization, mechanical checking, and open utilities also state management. however, numerous
WSN gadgets have serious asset demands as far as vitality, computation, and memory, brought
about by a need to cutoff the expense of the substantial number of gadgets needed for some
applications and by organization

Math Case Study
Problem 4. Prove that (y^3+z^3 ) x^2+yz^4 is irreducible over C[x,y,z]. Also prove that (y^3+z^3 )
x^2+y^2 z^3 is irreducible. Assume that (y^3+z^3 ) x^2+yz^4=a*b. Then one of a or b is linear in
x^2 and the other doesn't have x^2 at all because the degree of the product is the sum of the two
degrees. Now we write 〖a=cx〗^2+d, so c and d have only y's and z's. Then (y^3+z^3 )
x^2+yz^4=(〖cx〗^2+d)*b But now b*d=yz^4, and since C[y,z] is a unique factorization domain, b
and d must be monomials. But this means b is a monomial, in y and z. Since b*c=y^3+z^3 and we
see that y^3+z^3 factors to be (y+z)(y^2–yz+z^2). Since b is a monomial and b*c=(y+z)(y^2–
yz+z^2) Then b must be either (y+z) or (y^2–yz+z^2). However, they must not have ... Show more
Guess: Res(x–a,f(x))= b_n a^n+⋯+b_1 a+b_0. Lemma: det[■(0–a0@⋮⋮⋮@■(0@b_n
)■(0@b_(n–2) )■(0@b_(n–3) )) ■(...00@...⋮⋮@■(...@...)■(1@b_1 )■(–a@b_0
))]=b_n a^(n–1) Base Case: n = 2 det[■(0–a@b_2b_0 )]=0(–b_o )–b_2 (–a)= b_2 a n = 3
det[■(0–a0@01–a@b_3b_1b_0 )]=0–(–a)(0(b_0 )–b_3 (–a))=b_3 a^2 Induction:
Assume the determinant is b_n a^(n–1) is true for n≥3. Now look at n+1. det[■(0–
a0@⋮⋮⋮@■(0@b_(n+1) )■(0@b_(n–2) )■(0@b_(n–3) ))
■(...00@...⋮⋮@■(...@...)■(1@b_1 )■(–a@b_0 ))]=0–(–a)det[■(0–
a0@⋮⋮⋮@■(0@b_n )■(0@b_(n–2) )■(0@b_(n–3) ))
■(...00@...⋮⋮@■(...@...)■(1@b_1 )■(–a@b_0 ))] =a(b_(n+1) 〖(a〗^(n–1)))=b_(n+1)
a^n Proof: Base Case is n = 2: det[■(1–a0@01–a@b_2b_1b_0 )]=1*(1*(b_0 )–b_1*(–
a))–(–a)*(0*(b_0 )–b_2 (–a)) =b_0+b_1 a+b_2 a^2 Induction: Assume for n≥2, det[■(1–
a0@■(0@⋮)■(1@⋮)■(–a@⋮)@■(0@b_n )■(0@b_(n–1) )■(0@b_(n–2) ))
■(...00@■(...@...)■(0@⋮)■(0@⋮)@■(...@...)■(1@b_1 )■(–a@b_0 ))]=b_n a^n+⋯+b_1
a+b_0 Now look at n+1. det[■(1–a0@■(0@⋮)■(1@⋮)■(–a@⋮)@■(0@b_(n+1) )■(0@b_n
)■(0@b_(n–1) )) ■(...00@■(...@...)■(0@⋮)■(0@⋮)@■(...@...)■(1@b_1 )■(–a@b_0
))] =1*det[■(1–a0@■(0@⋮)■(1@⋮)■(–a@⋮)@■(0@b_n )■(0@b_(n–1) )■(0@b_(n–2) ))
■(...00@■(...@...)■(0@⋮)■(0@⋮)@■(...@...)■(1@b_1 )■(–a@b_0 ))]–(–a)det[■(0–
a0@■(0@⋮)■(1@⋮)■(–a@⋮)@■(0@b_(n+1) )■(0@b_(n–1) )■(0@b_(n–2) ))
■(...00@■(...@...)■(0@⋮)■(0@⋮)@■(...@...)■(1@b_1 )■(–a@b_0 ))] 〖=1*(b〗_n
a^n+〖=b〗_(n–1) a^(n–1)...+b_1 a+b_0+a(b_(n+1) a^n ) By

Volume Estimates of the Heart through the Use of Simpson's...
Table of Contents
1. Introduction and Objectives 3
2. Simpson's rule 3
2.1 First proof of Simpson's rule 4
2.2 Second proof of Simpson's rule 6
2.3 Error in Simpson's rule 7
2.4 Number of slices for the approximation to be exact up to a certain number 7
3. Application of the Simpson's rule to measuring the volume of the heart 8
3.3.2 Sample calculation. 10
4. Conclusion 11
5. Bibliography 12
6. Appendix 12
Introduction and Objectives
I was looking at a program on discovery channel where there were treasure hunter ships.They would
scan the seabed it. I noticed then when the seabed came up on the computer screen it was imaged
one part at a time, this made me think of integration with the trapezoidal rule where separate ...
Show more content on Helpwriting.net ...
Finally, I will draw the main conclusions from this project.
Simpson's rule
A way that we can see the trapezoidal (or midpoint) rule is that we approximate a function using a
first degree polynomial. Following this line of thinking we can say that by using Simpson's rule we
approximate a function using a second degree polynomial. In the graph here above f(x) represents
the function that we are trying to approximate
P(x) is the second degree polynomial that we use to approximate the function
The Simpson's rule can be derived in various ways described in the two following subsections.
First proof of Simpson's rule
(based on the proof in the book Calculus, Larson Hostetler Edwards's fifth edition)
∫_a^b▒█(f(x)dx=@ ) ∫_a^b▒〖(A〗 x^2+Bx+C)dx
Integral is taken
∫_a^b▒█(f(x)dx=@ ) [(Ax^3)/3+(Bx^2)/2+C] x = b – a
∫_a^b▒█(f(x)dx=[(A(b^3–a^3))/3+(B〖(b〗^2–a^2))/2+C(b–a)]@ )
By inverse distribution

∫_a^b▒f(x)dx=((b–a)/6)[2A(a^2+ab+b^2 )+3B(b+a)+6C]
∫_a^b▒f(x)dx=((b–a)/6)[(Aa^2+Ba+C)+4[A((b+a)/2)^2+B((b+a)/2)+C]+ (Ab^2+Bb+C)]
∫_a^b▒f(x)dx=((b–a)/6)[f(a)+4f((b+a)/2)+f(b)]
To use Simpson's rule you divide the interval into n equal subintervals each of width ∆x=(b–a)/n. N
is required to be even because for Simpson's rule we use coordinates to draw a polynomial of the
least degree to pass through. For a polynomial to be determined we need three coordinates, we reuse
the last coordinate of the previous polynomial to create our next one.

Identity Based Encryption Is A Critical Primitive Of...
Identity–Based Encryption Abdul Nayyer Mohammad Stephen Hyzny (Instructor) DATE:
11/29/2015 GOVERNORS STATE UNIVERSITY Abstract Identity based encryption is a critical
primitive of Identity based cryptography. All things considered it is a sort of open key encryption in
which people in public key of a client is some one of a kind data about the identity of the client.
Personality based encryption is an energizing distinct option for public key encryption, as this
encryption ends the essential requirement for a Public Key Infrastructure. Any setting, ... Show
more content on Helpwriting.net ...
Our plan expands on the thoughts of the Fuzzy IBE primitive and parallel tree information structure,
and is provably secure. Introduction Identity based encryption is an energizing distinct option for
open key encryption, which takes out the requirement for a Public Key Infrastructure (PKI) that
makes freely accessible the mapping between characters, public keys, and legitimacy of the latter.
The senders utilizing an IBE don 't have to look up in public keys and the relating declarations of
the collectors, in light of the fact that the personalities e.g. messages or IP addresses together with
regular open parameters are adequate for encryption. The private keys of the users are given by a
trustee outsider called the private key generator (PKG). As a solution for this issue for IBE, Boneh
and Franklin proposed that clients reestablish their private keys occasionally, e.g. consistently, and
senders utilize the receivers identities linked with the present time period. Notice that since just the
PKG 's open key and the recipient 's character are expected to encode, and there is no real way to
convey to the senders that a personality has been revoked, such a mechanism to routinely upgrade
clients private keys is by all accounts the main feasible answer for the renouncement issue. This
implies

Calorimetry Lab Report
With the ∆P calculated from equation 3, ∆P is converted to dynes/cm2. The values that were
calculated appear to be very similar to one another. The average ∆P are found to be
∆P= 4900(±200)
∆P= 4600 (±90)
for both sides of the capillaries. The values are indicated in the table below.
Table 1. Trials 1–5 indicate that one side of the capillary is tested. Trials 6–10 indicate the other side
of the capillary is being tested. Both radius values end up being very similar to one another, which
indicates that both sides are fire polished. The capillaries were tested using DI water in this scenario.
The values appeared to be very similar to one another. Values of ∆P are used in order to calculate the
radius of the capillary. The radius of the ... Show more content on Helpwriting.net ...
Indicates values for the concentration of the SDS solution made along with how it affected ∆P. The
SDS concentration is in terms of weight by volume and the radius is calculated from the previous
table.
With the following information a plot can be formed and the graph that is formed that it allows for a
polynomial graph to be present.
Figure 3. The equation for the formula is y = –1.36E+09x3 + 7.07E+06x2 – 1.38E+04x + 8.26E+01.
The error bars are indicated to ±1.9. The x–axis indicates the SDS concentration while the y axis
indicates γ which is the surface tension.
The rate of change of the equation of the line gives the value for dγ/(dc_s ) if a specific
concentration is plugged into it, which is then utilized in equation 2. In order to find the Γs
Table 3. Indicates the values for the R, dy/dcs for specific concentrations, which are indicated to the
right. Temperature is held constant in this scenario at 20˚C. The gas constant is in terms of
erg/K*mol.
With the data table, Γs vs. cs is graphed which includes the equation of the graph with the third
degree polynomial

The History Of Algebra, How It Started, The Most Study...
This paper will show the history of algebra, how it started, and how it grew to be what it is today. It
will show that it started it developments from the basic arithmetic operations that first were used to
solve simple addition, subtraction, multiplication, and division and how it went incorporating more
operations that permitted it to solve problems that involve abstract concepts. It will show that the
recorded history begins mostly with the Egyptian papyrus, and how it went passing from one
civilization. Moreover; it will show how each civilization contributed with something that at the end
helped it to become what it is today, the most study mathematical subject around the world.
1. Introductions
However; the recorded history begins with the Rhind Papyrus from Egypt and clay tablets from
Babylon, which describe the basic mathematical operations that include an unknown variable, then
it passed to India where written records in Sanskrit show that the Indian were already describing
verbally what seems to be the description of unknown variables. Next the recorded history passed to
the Greeks, where Diophantus published the b books that describe the syncopated algebra which are
considered to be the basis of all algebra used and studied by European mathematician.
Moreover; it was in Europe where the recorded history of the algebra really was taken to the next
level. The lead was taken by Fibonacci with is book liber abaci, who added the application of
sequences and crated a

Theory Of Computation : Assignment
NWODO NENNE 130805063 CSC 308 THEORY OF COMPUTATION ASSIGNMENT:
IDENTIFY 3 PROBLEMS THAT FALL INTO 1. CLASS P 2. CLASS NP DISCUSS IN DETAIL
ONE OF THE EXAMPLES IN EACH CLASS. DUE DATE: 14/09/2015 TO BE SUBMITTED ON
http://www.turnitin.com CLASS P COMPLEXITY P is the set of decidable languages which are in
polynomial time on a deterministic machine. Time complexity will be O(Nᵏ) where k is an integer.
Polynomial Time (P) – Any algorithm in class P can be solved in polynomial time – Any algorithm
running in polynomial time is said to be effective and labour–saving – If P can be solved in
polynomial time is denoted as a tractable decision problem Problems in Class P – Euler Paths/ Euler
Circuits – Searching – Single–source shortest path DETAILED DISCUSSION ON EUCLER
PATHS/ EUCLER CIRCUITS An Euler path is a path that uses every edge in a graph only once.
Euler paths do not have to return to the starting node/vertex. e.g An Euler circuit is a circuit that
uses all the edges in the graph exactly once. Due to the fact that it is a circuit, it has to start and end
at the same node. A graph is said to be an Eulerian if it has an Euler circuit, and semi Eulerian if it
has an Euler path. Euler Path/ Euler Circuit Theorems – A graph contains an Euler path iff (if and
only if) the graph contains exactly two nodes/vertices of odd degree. – A graph contains an Euler
circuit iff (if and only if) all nodes/vertices in the graph have even degree. – If none of these

Usepackage Essay
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usepackage[protrusion=true,expansion=true]{microtype}
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usepackage{graphicx}
usepackage{parallel,enumitem}
usepackage{amssymb}
usepackage{float}
usepackage{graphicx}
usepackage{rotating}
usepackage{amsmath, bm}
usepackage{amsfonts}
usepackage[T1]{fontenc}
usepackage[ polish,english]{babel}
usepackage[utf8]{inputenc}
usepackage{lmodern}
usepackage{subfigure} ... Show more content on Helpwriting.net ...
section{Vanilla options}
The following section illustrates the empirical results of the above–mentioned options pricing
techniques for one sample, consisting of 20 American put options, referred to as small sample.
Moreover, the comparison of the number of polynomial families, as well as the contrast between
regression methods are conducted.
section{Small sample results}
The analysis starts with the valuation of 20 American put options with the same set of parameters
usually considered in the literature, implemented by Longstaff and Schwartz as well. The following
Table 4.1 presents the results of pricing estimation of 20 options resulting from the combination of
the following parameters:
begin{gather*}
S(0) = { 36, 38, 40, 42, 44 }
sigma = { 0.2, 0.4 }
T = { 1, 2 }
end{gather*}
The strike price for all options was considered to be $K = 40$, the risk–free interest rate $r = 6%$

and the dividend yield $delta = 0$. The simulation was performed with 100,000 paths, with 50 time
steps per year and three weighted Laguerre

Wireless Sensor Networks ( Wsn ) Is A Key Innovation For...
Abstract–Wireless sensor networks (WSN) is a key innovation for the wireless network technology.
It generally has a large number of sensor nodes with a power unit, a sensing unit, a processing unit,
a storage unit, and a wireless transmitter receiver. They are more vulnerable to attack then wired
ones due to its nature and resources limitations. So as to overcome this security problems we can
different types of Key Distribution and Hierarchical WSN. This paper gives more solutions to all
problems in security issues of the Networks.
Keywords–wireless sensor network, distributed wireless sensor network, hierarchical wireless
sensor network, key distribution, key pre–distribution, pair–wise keys, group–wise key, network
keying.
I. ... Show more content on Helpwriting.net ...
(ii) Resource limitations on sensor nodes. (iii) Lack if fixed infrastructure. (iv) High risk of physical
attacks. (v) Unknown network topology prior to deployment. This is why it is tough to secure WSN.
II. TERMS, DEFINITIONS AND NOTATIONS
Terms used in this paper are as follows:
–key: symmetric key which is used to secure communication among two sensor nodes.
–pair–wise key: key which is used to unicast communication between a pair of sensor nodes.
–group–wise key: key which is used to secure multicast communication among a group of sensor
nodes .
–network–wise key: key which is used to secure broadcast messages.
III. NETWORK MODELS AND SECURITY REQUIREMENTS OF WSN
There are generally 2 types of architectures in WSNs. (i) Hierarchical and (ii) Distributed. In a
Hierarchical WSN, A base station is a gateway to another network with some other base station to a
data processing and storage center, or a human interface node point. Base point takes data from
sensors send it to processing unit and that data is sent to the human interface node point. Therefore,
base stations are used as key distribution centers as they are connected every other node in network.
Sensor nodes form a dense network in form of clusters where a cluster of sensors lying in a specific
area may provide similar or close readings of data. Not all node gets the power from power unit few
sensor nodes depend on the ad hoc communication to reach base stations.

Power Series Method For Solving Linear Differential Equations
ABSTRACT In this work, we studied that Power Series Method is the standard basic method for
solving linear differential equations with variable coefficients. The solutions usually take the form
of power series; this explains the name Power series method. We review some special second order
ordinary differential equations. Power series Method is described at ordinary points as well as at
singular points (which can be removed called Frobenius Method) of differential equations. We
present a few examples on this method by solving special second order ordinary differential
equations.
Key words ; Power series, differential equations, Frobenius Method, Lengendre polynomials
1.0 INTRODUCTION
1.1 BACKGROUND OF THE STUDY
The attempt to solve physical problems led gradually to Mathematical models involving an equation
in which a function and its derivatives play important role. However, the theoretical development of
this new branch of Mathematics –Differential Equations– has its origin rooted in a small number of
Mathematical problems. These problems and their solutions led to an independent discipline with
the solution of such equations an end in itself (Sasser, 2005).
1.2 STATEMENT OF THE PROBLEM
The research work seeks to find solutions of second–order ordinary differential equations using the
power series method.
1.3 AIM AND OBJECTIVES
The aim and objectives of the study are to: Describe the power series method. Use it to solve linear
ordinary differential equations with

Mayan Math History
Like each other part of human advancement, mathematics has its own particular birthplace focused
around the needs of humanity in searching for understanding. Mathematics emerged from the
necessity to quantify time and number. The earliest evidence of counting occurred in mountains of
Africa were notched bones and scored pieces of wood and stone were discovered. As human
advancements started to surface in Asia and the near east, frameworks and essential appreciation of
arithmetic, geometry and polynomial math started to develop. Mathematics has made a lot of
progress from the first evidence of counting in 50,00 B.C to the current utilization of math all over
the place from cellphones and machines to dating of old ancient rarities and adjusting ... Show more
The importance of astronomy and calendar (18 months a year; 20 days a month) calculations to the
Mayans required mathematics, which lead them create an advanced number system of its time. The
system was on base 20 and most times base 5, which very likely originated from counting on fingers
and toes. The numerals consisted of only three symbols: zero was represented as a shell shape, one,
a dot and five a bar, so calculation was basically adding or subtracting bars and dots. In spite of not
having the idea of a division, they delivered greatly exact cosmic perceptions utilizing no
instruments other than sticks in form of crosses to view astronomical objects and had the capacity
measure the length of the sunlight based year to a far higher level of exactness than that utilized as a
part of Europe (their computations created 365.242 days, contrasted with the cutting edge estimation
of 365.242198), and in addition the length of the lunar month (their evaluation was 29.5308 days,
contrasted with the advanced estimation of 29.53059). Due to the geographic disconnect, Mayan
mathematics hardly influenced the Old World (Europe and Asia) mathematics and numbering

Analytical Mathematics: Matrices
Analytical Mathematics: Matrices Matrices Matrices are a grouping of numbers or symbols that
represent numbers in a semi–rectangle that are placed into order through one or a series of vertical
lines known as columns and one or a series of horizontal lines known as rows. Matrices come in
many sizes, they can be as few as one number and could go all the way to infinity. Not just one
number or letter can be placed in a space a whole equation with numbers and variables can be
placed in a single space. A matrix by itself though is useless, but when a number or equation is
placed outside of it can now be multiplied. If a matrix is placed next to another matrix then the
possibilities and problems that can be solved are almost endless of what can be accomplished. A
matrix can be added, subtracted, multiplied, divided, and can have the inverse taken out of it. When
a matrix is placed parallel to another matrix separated by a plus symbol, one very important thing
must be taken into consideration. Which is do the matrices being added together have the same
number of rows and columns, for example, matrix one is a three by three which means it has three
rows and three columns and matrix two is also a three by three. The first step is to add up the first
row in matrix one with the first column in row two and then again for the second row and column.
After those steps are followed the addition of the matrix should be complete. In subtraction of
matrices the same steps should be

Niels Abel Research Paper
Niels Abel and some of his work Many innovational mathematicians come and go, but only a few
remembered for their great accomplishments. Niels Henrik Abel is one of the greatest
mathematicians that have influenced modern mathematics, solving and creating theorems, like the
Abelian–Ruffini theorem and Abel's theorem, and formulas/equations, like the abel equation, Abel's
inequality. He started discovering and creating these at a young age. Niels Abel was born in Norway,
in a neighborhood parish, to Georg Abel, who was a pastor with a degree in theology and
philosophy, and Anne Marie Simonsen, a daughter to a wealthy merchant. Abel was homeschooled
by his father and was giving hand written books to study from. When Norway gained its
independence, Georg Abel was elected to the Norwegian legislature, also known as the Storting, and
George decided to send one of his sons to the cathedral school, at which Georg worked. The
following year he sent his eldest son to the school. this was the first time Niels ever went to school.
Niels started cathedral school at age 13 and was shared rooms and classes with older brothers.
compared to his older brother, Niels didn't seem to be a prodigy because his brothers grades were far
better than his own. Only when a new teacher, Bernt Holmboe, started at the school, did Niels begin
to take interests in math and caught the attention of the new math teacher. Bernt began to give Niels
higher level math and give him extra tutoring. after

The Low Form Credit Risk Model
Specification of the model.
In this dissertation the focus is on the reduced form credit risk model to calculate the prices of the
default–free zero–coupon bonds. In order to use this model the specification of the models'
parameters should be done. More precisely, the hazard rate risk–neutral model, recovery rate and a
default–free term structure should be chosen.
Hazard process.
There are several most popular approaches for hazard process estimation.
First approach defines the hazard process as a stochastic process with help of Vasicek and CIR
processes. Another approach that underlies the cross–sectional estimation considers the hazard
process to be either a stochastic or a constant. When the hazard rate is a stochastic process, there ...
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First way is to estimate it is a parameter from a real market data and second way is to assume this
rate as a constant. From the first point of view it seems that the first method is more preferable than
the second one, it is difficult and consuming in implementation.
The following graph shows a model obtained from the following security: a bond issued by
Deutsche Bank that was taken from Bloomberg on the 23d of July. The swap curve was used in the
following graphs as a proxy for the default–free curve for a constant hazard rate model. The curves
are built for a constant recovery rates that vary from 10%–90% with a step of 10%. For each
obtained value the hazard rate is estimated.
Graph 1. Graph one represents the fitted zero–coupon curve obtained from the Deutsche Bank bond
from the bid quotes. The vertical axes is the zero–coupon rate and the horizontal axes is time to
maturity. The curve was calculated by Nelson–Siegel method in Matlab (see Appendix 1).
Graph 2. The second graph is function of 1 – recovery rate where recovery rate changes from 10%
to 90%. The vertical axes values of the function one minus recovery rate, meanwhile the horizontal
axes shows the values of the recovery rate.
Graph 3. The third graph shows the hazard rate function that, as it was mentioned earlier, is a
polynomial function of time to maturity and parameters λ_i for i=1,...,n. The horizontal axes shows
the recovery rate values, meanwhile the

Julius Caesar Galois Research Paper
Galois was born into a family involved in the French First Republic in Bourg–la– Reine a commune
in the French Empire. His father was amiable and highly philosophical in nature;he directed a
school educating about sixty boarders. He was later elected mayor of Bourg–la–Reine. His mother,
Adelaïde–Marie, was from a family of jurists and had received a more traditional education; she was
well–versed in Latin and Classical Literature. She had a headstrong personality and was eccentric,
even considered queer by many. Galois had a sister named Nathalie–Théodore and a brother called
Alfred. Évariste received his primary education from his mother. She looked forward to inculcate in
him, along with the elements of classical culture and Literature, ... Show more content on
Helpwriting.net ...
Although he was unable to verbally explain himself Galois preformed and revolutionized people's
views on mathematics with his new ideas on radicals and degree which many at the time had never
thought of. Through the struggling times of France, Galois was able to live out his (short) life and
make an impact on the world. Most of Galois' work was never accepted much less even appreciated
by society in the era but after his death professors were able to decipher his methods which could've
been done much quicker had he been able to explain himself. To this day Evariste Galois is seen as a
master mathematician and although his grave remains nameless and the location of that grave
unknown many can thank him for his contributions to the culture and the

Charles Babbage, A Brilliant And Well Educated Man
Charles Babbage was a brilliant and well educated man. His mother, Plumleigh Babbage gave birth
to him in 1791 A.D. His father's name was Benjamin Babbage. He was one of four children. His
father was a banking partner of founding Praed's Co. in London. Benjamin was a rich man, so he
and his wife had big plans for Charles to attend many good schools; but when he was around the age
of eight, he had to move to a country school in order to recover from a life threatening fever. He had
an unstable and delicate health status for most of his early life, which kept him from going to the
schools his parents wanted him to go to. He relied on private lessons and teachings for most of his
childhood. He attended two different colleges as well. First, he went to Trinity College in
Cambridge but wasn't pleased with the math program there. So, he then attended Peterhouse, in
Cambridge. He was the best mathematician at Peterhouse. He graduated from Peterhouse but didn't
receive an honorary degree. He wasn't successful when he first graduated. He did some lectures on
astronomy every now and then; And eventually helped to establish the Astronomical Society. A little
later he received his honorary degree without question. After receiving his honorary degree, he
married Georgiana Whitmore. They had eight children together. Five of their children never lived to
become adults. In 1822, Charles designed the plan and idea of a machine that he called the
difference machine. The

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