Department of Mathematics MTL107: Numerical Methods and Computations Exercise Set 8: Approximation-Linear Least Squares Polynomial approximation, Chebyshev Polynomial approximation. 1. Compute the linear least square polynomial for the data: i xi yi 1 0 1.0000 2 0.25 1.2840 3 0.50 1.6487 4 0.75 2.1170 5 1.00 2.7183 2. Find the least square polynomials of degrees 1,2 and 3 for the data in the following talbe. Compute the error E in each case. Graph the data and the polynomials. : xi 1.0 1.1 1.3 1.5 1.9 2.1 yi 1.84 1.96 2.21 2.45 2.94 3.18 3. Given the data: xi 4.0 4.2 4.5 4.7 5.1 5.5 5.9 6.3 6.8 7.1 yi 113.18 113.18 130.11 142.05 167.53 195.14 224.87 256.73 299.50 326.72 a. Construct the least squared polynomial of degree 1, and compute the error. b. Construct the least squared polynomial of degree 2, and compute the error. c. Construct the least squared polynomial of degree 3, and compute the error. d. Construct the least squares approximation of the form beax, and compute the error. e. Construct the least squares approximation of the form bxa, and compute the error. 4. The following table lists the college grade-point averages of 20 mathematics and computer science majors, together with the scores that these students received on the mathematics portion of the ACT (Americal College Testing Program) test while in high school. Plot these data, and find the equation of the least squares line for this data: : ACT Grade-point ACT Grade-point score average score average 28 3.84 29 3.75 25 3.21 28 3.65 28 3.23 27 3.87 27 3.63 29 3.75 28 3.75 21 1.66 33 3.20 28 3.12 28 3.41 28 2.96 29 3.38 26 2.92 23 3.53 30 3.10 27 2.03 24 2.81 5. Find the linear least squares polynomial approximation to f(x) on the indicated interval if a. f(x) = x2 + 3x+ 2, [0, 1]; b. f(x) = x3, [0, 2]; c. f(x) = 1 x , [1, 3]; d. f(x) = ex, [0, 2]; e. f(x) = 1 2 cosx+ 1 3 sin 2x, [0, 1]; f. f(x) = x lnx, [1, 3]; 6. Find the least square polynomial approximation of degrees 2 to the functions and intervals in Exercise 5. 7. Compute the error E for the approximations in Exercise 6. 8. Use the Gram-Schmidt process to construct φ0(x), φ1(x), φ2(x) and φ3(x) for the following intervals. a. [0,1] b. [0,2] c. [1,3] 9. Obtain the least square approximation polynomial of degree 3 for the functions in Exercise 5 using the results of Exercise 8. 10. Use the Gram-Schmidt procedure to calculate L1, L2, L3 where {L0(x), L1(x), L2(x), L3(x)} is an orthogonal set of polynomials on (0,∞) with respect to the weight functions w(x) = e−x and L0(x) = 1. The polynomials obtained from this procedure are called the La- guerre polynomials. 11. Use the zeros of T̃3, to construct an interpolating polynomial of degree 2 for the following functions on the interval [-1,1]: a. f(x) = ex, b. f(x) = sinx, c. f(x) = ln(x+ 2), d. f(x) = x4. 12. Find a bound for the maximum error of the approximation in Exercise 1 on the interval [-1,1]. 13. Use the zer.

1. Department of Mathematics
MTL107: Numerical Methods and Computations
Exercise Set 8: Approximation-Linear Least Squares Polynomial
approximation, Chebyshev
Polynomial approximation.
1. Compute the linear least square polynomial for the data:
i xi yi
1 0 1.0000
2 0.25 1.2840
3 0.50 1.6487
4 0.75 2.1170
5 1.00 2.7183
2. Find the least square polynomials of degrees 1,2 and 3 for the
data in the following talbe.
Compute the error E in each case. Graph the data and the
polynomials.
:
xi 1.0 1.1 1.3 1.5 1.9 2.1
yi 1.84 1.96 2.21 2.45 2.94 3.18

2. 3. Given the data:
xi 4.0 4.2 4.5 4.7 5.1 5.5 5.9 6.3 6.8 7.1
yi 113.18 113.18 130.11 142.05 167.53 195.14 224.87 256.73
299.50 326.72
a. Construct the least squared polynomial of degree 1, and
compute the error.
b. Construct the least squared polynomial of degree 2, and
compute the error.
c. Construct the least squared polynomial of degree 3, and
compute the error.
d. Construct the least squares approximation of the form beax,
and compute the error.
e. Construct the least squares approximation of the form bxa,
and compute the error.
4. The following table lists the college grade-point averages of
20 mathematics and computer
science majors, together with the scores that these students
received on the mathematics
portion of the ACT (Americal College Testing Program) test
while in high school. Plot
these data, and find the equation of the least squares line for
this data:

3. :
ACT Grade-point ACT Grade-point
score average score average
28 3.84 29 3.75
25 3.21 28 3.65
28 3.23 27 3.87
27 3.63 29 3.75
28 3.75 21 1.66
33 3.20 28 3.12
28 3.41 28 2.96
29 3.38 26 2.92
23 3.53 30 3.10
27 2.03 24 2.81
5. Find the linear least squares polynomial approximation to
f(x) on the indicated interval
if
a. f(x) = x2 + 3x+ 2, [0, 1]; b. f(x) = x3, [0, 2];
c. f(x) = 1
x

4. , [1, 3]; d. f(x) = ex, [0, 2];
e. f(x) = 1
2
cosx+ 1
3
sin 2x, [0, 1]; f. f(x) = x lnx, [1, 3];
6. Find the least square polynomial approximation of degrees 2
to the functions and intervals
in Exercise 5.
7. Compute the error E for the approximations in Exercise 6.
8. Use the Gram-Schmidt process to construct φ0(x), φ1(x),
φ2(x) and φ3(x) for the following
intervals.
a. [0,1] b. [0,2] c. [1,3]
9. Obtain the least square approximation polynomial of degree 3
for the functions in Exercise
5 using the results of Exercise 8.
10. Use the Gram-Schmidt procedure to calculate L1, L2, L3
where {L0(x), L1(x), L2(x), L3(x)}
is an orthogonal set of polynomials on (0,∞) with respect to the
weight functions w(x) =
e−x and L0(x) = 1. The polynomials obtained from this
procedure are called the La-

5. guerre polynomials.
11. Use the zeros of T̃3, to construct an interpolating
polynomial of degree 2 for the following
functions on the interval [-1,1]:
a. f(x) = ex, b. f(x) = sinx, c. f(x) = ln(x+ 2), d. f(x) = x4.
12. Find a bound for the maximum error of the approximation in
Exercise 1 on the interval
[-1,1].
13. Use the zeros of T̃3 and transformations of the given
interval to construct an interpolating
polynomial of degree 2 for the following functions on the
indicated intervals:
a. f(x) = 1
x
, [1, 3]; b. f(x) = e−x, [0, 2];
c. f(x) = 1
2
cosx+ 1
3
sin 2x, [0, 1]; d. f(x) = x lnx, [1, 3];
14. Find the sixth Maclaurin polynomial for sinx, and use
Chebyshev economization to obtain

6. a lesser degree polynomial approximation while keeping the
error less than 0.01 on [-1,1].
15. Show that for each Chebyshev polynomial Tn(x), we have∫ 1
−1
[Tn(x)]
2
√
1− x2
dx =
π
2
.
16. Show that for each n, the derivative of the Chebyshev
polynomial Tn(x), has n−1 distinct
zeros in (-1,1).
ANSWERS
1. The linear least-squares polynomial is 1.70784x+ 0.89968.
2. The least-squares polynomials with their errors are,
respectively, 0.6208950 + 1.219621x,
with E = 2.719× 10−5;
0.5965807 + 1.253293x− 0.01085343x2, with E = 1.801× 10−5;
and
0.6290193 + 1.185010x+ 0.03533252x2 − 0.01004723x3, with E
= 1.741× 10−5.

7. 3. a. The linear least-squares polynomial is 72.0845x− 194.138,
with error 329.
b. The least-squares polynomial of degree two is 6.61821x2 −
1.14352x + 1.23556, with
error 1.44× 10−3.
c. The least-squares polynomial of degree three is
−0.0136742x3+6.84557x2−2.37919x+
3.42904, with error 5.27× 10−4.
d. The least-squares polynomial of the form beax is
24.2588e0.372382x, with error 418.
e. The least-squares polynomial of the form bxa is
6.23903x2.01954, with error 0.00703.
4. The least squares line for the point average is 0.101 (ACT
score) +0.487
5. The linear least-sqaures approximations are:
a. P1(x) = 1.833333 + 4x b. P1(x) = −1.600003 + 3.600003x
c. P1(x) = 1.140981− 0.2958375x d. P1(x) = 0.1945267 +
3.000001x
e. P1(x) = 0.6109245 + 0.09167105x f. P1(x) = −1.861455 +
1.666667x.
6. The linear least-sqaures approximations of degree two are:
a. P2(x) = 2.000002 + 2.999991x+ 1.000009x
2
b. P2(x) = 0.4000163− 2.400054x+ 3.000028x2
c. P2(x) = 1.723551− 0.9313682x+ 0.1588827x2
d. P2(x) = 1.167179 + 0.08204442x+ 1.458979x

8. 2
e. P2(x) = 0.4880058 + 0.8291830x− 0.7375119x2
f. P2(x) = −0.9089523 + 0.6275723x+ 0.2597736x2.
7. a. 0.3427× 10−9 b. 0.0457142 c. 0.000358354
d. 0.0106445 e. 0.0000134621 f. 0.00000967795.
8. The Gram-Schmidt process produces the following
collections of polynomials:
a. φ0(x) = 1, φ1(x) = x− 0.5, φ2(x) = x2 − x+ 16 , and φ3(x) = x
3 − 1.5x2 + 0.6x− 0.05
b. φ0(x) = 1, φ1(x) = x− 1, φ2(x) = x2 − 2x+ 23 , and φ3(x) = x
3 − 3x2 + 12
5
x− 2
5
c. φ0(x) = 1, φ1(x) = x− 2, φ2(x) = x2 − 4x+ 113 , and φ3(x) = x
3 − 6x2 + 11.4x− 6.8
9. The least-squares polynomial of degree two are:
a. P2(x) = 3.833333φ0(x) + 4φ1(x) + 0.9999998φ2(x),
b P2(x) = 2φ0(x) + 3.6φ1(x) + 3φ2(x),
c. P2(x) = 0.5493061φ0(x)− 0.2958369φ1(x) + 0.1588785φ2(x),
d. P2(x) = 3.194528φ0(x) + 3φ1(x) + 1.458960φ2(x),
e. P2(x) = 0.6567600φ0(x) + 0.09167105φ1(x)−
0.73751218φ2(x),

9. f. P2(x) = 1.471878φ0(x) + 1.666667φ1(x) + 0.2597705φ2(x).
10. The Laguerre polynomials are L1(x) = x− 1, L1(x) = x2 −
4x+ 2, and
L3(x) = x
3 − 9x2 + 18x− 6.
11. The interpolating polynomials of degree two are:
a. P2(x) = 2.377443 + 1.590534(x− 0.8660254) +
0.5320418x(x− 0.8660254)
b. P2(x) = 0.7617600 + 0.8796047(x− 0.8660254)
c. P2(x) = 1.052926 + 0.4154370(x− 0.8660254)−
0.1384262x(x− 0.8660254)
d. P2(x) = 0.5625 + 0.64519(x− 0.8660254) + 0.75x(x−
0.8660254)
12. Bounds for the maximum errors of polynomials in Exercise
11 are:
a.0.1132617 b. 0.04166667 c. 0.08333333 d. 1.000000
13. The zeros of T̃3 produce the following interpolating
polynomials of degree two:
a. P2(x) = 0.3489153− 0.1744576(x− 2.866025) + 0.1538462(x−
2.866025)(x− 2)
b. P2(x) = 0.1547375− 0.2461152(x− 1.866025) + 0.1957273(x−
1.866025)(x− 1)
c. P2(x) = 0.6166200− 0.2370869(x− 0.9330127)−
0.7427732(x− 0.9330127)(x− 0.5)
d. P2(x) = 3.0177125 + 1.883800(x− 2.866025) + 0.2584625(x−

10. 2.866025)(x− 2)
14. The cubic polynomial 383
384
x− 5
32
x3 approximates sinx with error at most 7.19× 10−4.
15. The change of variable x = cos θ produces∫ 1
−1
T 2n(x)√
1−x2dx =
∫ 1
−1
[cos(n arccosx)]2√
1−x2 dx =
∫ π
0
(cos(nθ))2dθ = π
2
.
16. It was shown in text that the zeros of T
′
n(x) occur at x
′
k = cos(kπ/n) for k = 1, .., n− 1.
Because x

11. ′
0 = cos(0), x
′
n = cos(π/) = −1, and all values of the cosine lie in the interval
[-1,1] it reamins only to show that the zeros are distinct. This
follows from the fact
that for each k = 1, .., n − 1, we have x′k in the interval (0, π)
and on this interval
Dx cos(x) = − sinx < 0. As a consequence, T
′
n(x) is one-to-one on (0, π), and these n− 1
zeros of T
′
n(x) are distinct.
Syllabus for APS-295
ASSOCIATE CAPSTONE
COURSE DESCRIPTION
The Associate Capstone prepares and develops students’ skills
for a technical workstream leader role in their area of discipline
within business and technology. This course teaches various
techniques to simulate new concepts for a technology driven
ideation process and the ability to assess the marketplace.
Throughout this course the students will develop their ability
to: understand and manage technology lifecycles; recognize
business and manufacturing tools and strategies to yield the
greatest efficiencies through each stage of this process; and to
anticipate the issues and considerations when deploying

12. technology. This course is designed to provide knowledge in
these areas for the identification, analysis and synthesis of
current trends and incremental changes in any technical area of
study.
COURSE OBJECTIVES
After completing this course, you should be able to:
· CO1 Identify the KSA’s (knowledge, skills and abilities)
needed for a technical workstream lead role in the applied
science and technology discipline.
· CO2 Describe technical innovation objectives, processes,
techniques, and outcomes for new applications.
· CO3 Explain the success of technology deployment in the
industry through market potential, segmentation, differentiation
and trends.
· CO4 Manage, protect and increase technology lifecycles of
applications.
· CO5 Describe common engineering and manufacturing
processes to establish baselines, eliminate waste, and identify
improvement opportunities for technology development.
· CO6 Discuss the connection between science and technology
when developing new solutions or applications.
· CO7 Summarize how commercialized technologies can have
large scale implications across the economy, medical and
political communities.
· CO8 Apply approaches to identify gaps in technology and
anticipate challenges through the application lifecycle.
· CO9 Review factors that influence opinions and perceptions of
new technology applications.
COURSE MATERIALS
There are no textbooks required for this course.
Web links to recommended readings are included in each
module. You should also search journal articles through
theresearch databases for your final project. You can find links
to the EBSCOhost and ProQuest databases by logging into the
myEdison portal and locating the My Resources section in the
center of the page. The links can be found under the

13. Educational tab.
COURSE STRUCTURE
Associate Capstone is a three-credit online course, consisting of
six modules. Modules include topics, learning objectives, study
materials, and activities. Module titles are listed below.
· Module 1: Technology Introduction
· Module 2: Ideation
· Module 3: The Marketplace
· Module 4: Technology Lifecycle
· Module 5: Engineering and Manufacturing Processes
· Module 6: Deployment Considerations
BEFORE YOU START YOUR RESEARCH
One or more of the assignments in this course may involve
original research. Research on persons other than yourself may
require approval by the Institutional Review Board (IRB) of
Thomas Edison State University prior to beginning your
research. Examples of research types that may need IRB review
are questionnaires, surveys, passive observation of individuals,
interviews, and experimental procedures. Research involving
vulnerable populations will always need IRB review. An IRB
review is designed to protect research subjects from potential
harm.
The following links fully explain the purpose of the
Institutional Research Board as well as how to determine if your
research requires IRB review. If you are in doubt, always ask
for guidance from the University.
· Institutional Review Board (general)
· Types of IRB Review
· IRB Forms
· Policies and Procedures
· FAQs and Resources
ASSESSMENT METHODS
For your formal work in the course, you are required to
participate in online discussions, complete written assignments,
and finish a final project. See below for more details.
Consult the Course Calendar for assignment due dates.

14. Discussion Forums
In addition to an ungraded Introductions Forum in Module 1,
Associate Capstone requires you to participate in six graded
online discussion forums. For this course, you are required to
make a minimum of four comments on the responses of at least
two other classmates. Your initial posting should include more
than three resources, which must be referenced using APA style.
Communication with the mentor and among fellow students is a
critical component of online learning. Participation in online
discussions involves two distinct assignments: an initial
response to a posted question and subsequent comments on
classmates' responses.
You will be evaluated both on the quality of your responses
(i.e., your understanding of readings and concepts as
demonstrated by well-articulated, critical thinking) and quantity
of your participation (i.e., the number of times you participate
meaningfully in the assigned forums). Responses and comments
should be properly proofread and edited, professional, and
respectful.
Meaningful participation in online discussions is relevant to the
content, adds value, and advances the discussion. Comments
such as "I agree" and "ditto" are not considered value-adding
participation. Therefore, when you agree or disagree with a
classmate, the reading, or your mentor, state and support your
agreement or disagreement.
For posting guidelines and help with discussion forums, please
see the Student Handbook located within the General
Information page of the course Web site. Click to view Online
Discussion Grading Rubric.
Written Assignments
You are required to complete six written assignments. The
written assignments consist of specific topics defined in each
course module focusing on the areas below:
· Ideation
· Marketplace
· Lifecycle

15. · Engineering and Manufacturing Considerations
· Deployment
This expertise is vital when developing and commercializing
new applications. The course consists of readings and
discussions in general science and technology issues and trends,
and the application of the material to specific areas of
technology. The latter is accomplished via the development of a
final project by each student on a new technology area within
their discipline. Each written assignment is structured to build
upon the previous ones. The final project will contain the four
written assignments combined as an integrated document
including all of the changes made based on the mentor’s
feedback.
For help regarding preparing and submitting assignments, see
the Student Handbook located within the General Information
page of the course Web site.
· Click to review the grading rubric for written assignments.
Final Project
There will be a final project as stated in Written
Assignments section. You may check Module 6 for more details
about the final project. There are no quizzes or examinations in
this course.
GRADING AND EVALUATION
Your grade in the course will be determined as follows:
· Online discussion (6)—30 percent
· Written assignments (5)—50 percent
· Final Project/Written Assignment 6—20 percent
All activities will receive a numerical grade of 0–100. You will
receive a score of 0 for any work not submitted. Your final
grade in the course will be a letter grade. Letter grade
equivalents for numerical grades are as follows:
A
=
93–100
C+

16. =
78–79
A–
=
90–92
C
=
73–77
B+
=
88–89
C–
=
70–72
B
=
83–87
D
=
60–69
B–
=
80–82
F
=
Below 60
To receive credit for the course, you must earn a letter grade of
C or better (for an area of study course) or D or better (for a
course not in your area of study), based on the weighted average
of all assigned course work (e.g., exams, assignments,
discussion postings, etc.).
STRATEGIES FOR SUCCESS

17. First Steps to Success
To succeed in this course, take the following first steps:
· Read carefully the entire Syllabus, making sure that all aspects
of the course are clear to you and that you have all the materials
required for the course.
· Take the time to read the entire Online Student Handbook. The
Handbook answers many questions about how to proceed
through the course, how to schedule exams, and how to get the
most from your educational experience at Thomas Edison State
University.
· Arrange to take your examinations by following the
instructions in this Syllabus and the Online Student Handbook.
· Familiarize yourself with the learning management systems
environment—how to navigate it and what the various course
areas contain. If you know what to expect as you navigate the
course, you can better pace yourself and complete the work on
time.
· If you are not familiar with Web-based learning be sure to
review the processes for posting responses online and
submitting assignments before class begins.
Study Tips
Consider the following study tips for success:
· To stay on track throughout the course, begin each week by
consulting the course Calendar. The Calendar provides an
overview of the course and indicates due dates for submitting
assignments, posting discussions, and scheduling and taking
examinations.
· Check Announcements regularly for new course information.
ACADEMIC INTEGRITY
Thomas Edison State University is committed to maintaining
academic quality, excellence, and honesty. The University
expects all members of its community to share the commitment
to academic integrity, an essential component of a quality
academic experience.
Students at Thomas Edison State University are expected to
exhibit the highest level of academic citizenship. In particular,

18. students are expected to read and follow all policies,
procedures, and program information guidelines contained in
publications; pursue their learning goals with honesty and
integrity; demonstrate that they are progressing satisfactorily
and in a timely fashion by meeting course deadlines and
following outlined procedures; observe a code of mutual respect
in dealing with mentors, staff, and other students; behave in a
manner consistent with the standards and codes of the
profession in which they are practicing; keep official records
updated regarding changes in name, address, telephone number,
or e-mail address; and meet financial obligations in a timely
manner. Students not practicing good academic citizenship may
be subject to disciplinary action including suspension,
dismissal, or financial holds on records.
All members of the University community are responsible for
reviewing theAcademic Code of Conduct Policy in the
University Catalog and online at www.tesu.edu.
Academic Dishonesty
Thomas Edison State University expects all of its students to
approach their education with academic integrity—the pursuit
of scholarly activity free from fraud and deception. All mentors
and administrative staff members at the University insist on
strict standards of academic honesty in all courses. Academic
dishonesty undermines this objective. Academic dishonesty can
take the following forms:
· Cheating
· Gaining or providing unauthorized access to examinations or
using unauthorized materials during exam administration
· Submitting credentials that are false or altered in any way
· Plagiarizing (including copying and pasting from the Internet
without using quotation marks and without acknowledging
sources)
· Forgery, fabricating information or citations, or falsifying
documents
· Submitting the work of another person in whole or in part as
your own (including work obtained through document sharing

19. sites, tutoring schools, term paper companies, or other sources)
· Submitting your own previously used assignments without
prior permission from the mentor
· Facilitating acts of dishonesty by others (including making
tests, papers, and other course assignments available to other
students, either directly or through document sharing sites,
tutoring schools, term paper companies, or other sources)
· Tampering with the academic work of other students
Plagiarism
Thomas Edison State University is committed to helping
students understand the seriousness of plagiarism, which is
defined as using the work and ideas of others without proper
citation. The University takes a strong stance against
plagiarism, and students found to be plagiarizing are subject to
discipline under the academic code of conduct policy.
If you copy phrases, sentences, paragraphs, or whole documents
word-for-word—or if you paraphrase by changing a word here
and there—without identifying the author, or without
identifying it as a direct quote, then you are plagiarizing. Please
keep in mind that this type of identification applies to Internet
sources as well as to print-based sources. Copying and pasting
from the Internet, without using quotation marks and without
acknowledging sources, constitutes plagiarism. (For information
about how to cite Internet sources, see Online Student
Handbook > Academic Standards > “Citing Sources.”)
Accidentally copying the words and ideas of another writer does
not excuse the charge of plagiarism. It is easy to jot down notes
and ideas from many sources and then write your own paper
without knowing which words are your own and which are
someone else’s. It is more difficult to keep track of each and
every source. However, the conscientious writer who wishes to
avoid plagiarizing never fails to keep careful track of sources.
Always be aware that if you write without acknowledging the
sources of your ideas, you run the risk of being charged with
plagiarism.
Clearly, plagiarism, no matter the degree of intent to deceive,

20. defeats the purpose of education. If you plagiarize deliberately,
you are not educating yourself, and you are wasting your time
on courses meant to improve your skills. If you plagiarize
through carelessness, you are deceiving yourself.
For examples of unintentional plagiarism, advice on when to
quote and when to paraphrase, and information about writing
assistance and originality report checking, click the links
provided below.
Examples of Unintentional Plagiarism
When to Quote and When to Paraphrase
Writing Assistance at Smarthinking
Originality Report Checking at Turnitin
Disciplinary Process for Plagiarism
Acts of both intentional and unintentional plagiarism violate
theAcademic Code of Conduct.
If an incident of plagiarism is an isolated minor oversight or an
obvious result of ignorance of proper citation requirements, the
mentor may handle the matter as a learning exercise.
Appropriate consequences may include the completion of
tutorials, assignment rewrites, or any other reasonable learning
tool in addition to a lower grade for the assignment or course.
The mentor will notify the student and appropriate dean of the
consequence by e-mail.
If the plagiarism appears intentional and/or is more than an
isolated incident, the mentor will refer the matter to the
appropriate dean, who will gather information about the
violation(s) from the mentor and student, as necessary. The
dean will review the matter and notify the student in writing of
the specifics of the charge and the sanction to be imposed.
Possible sanctions include:
· Lower or failing grade for an assignment
· Lower or failing grade for the course
· Rescinding credits
· Rescinding certificates or degrees
· Recording academic sanctions on the transcript
· Suspension from the University

21. · Dismissal from the University
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