find six distinct roots of unity which sum to zero but for which no nonempty proper subset sums to zero
Answers
No nonempty proper subset of these roots sums to zero.
The six distinct roots of unity that sum to zero but for which no nonempty proper subset sums to zero are:
\(1, w, w^2, -1, -w, -w^2\)
where w is a complex cube root of unity, i.e., \(w^3 = 1.\)
To see that these roots sum to zero, we can use the fact that the sum of all nth roots of unity is zero, except for the case n = 1, where the sum is 1. That is,
\(1 + w + w^2 + ... + w^{(n-1)} = 0\)
if n > 1, and
1 = 1
if n = 1.
For the case n = 6, we have the six distinct roots of unity \(1, w, w^2, -1, -w, -w^2\), and their sum is:
\(1 + w + w^2 + (-1) + (-w) + (-w^2) = 0\)
To see that no nonempty proper subset of these roots sums to zero, we can use a proof by contradiction. Suppose there exists a nonempty proper subset S of these roots such that the sum of the elements in S is zero. Since S is nonempty, it must contain at least one of the roots 1, w, or \(w^2\). Without loss of generality, assume that S contains the root 1. Then, S must also contain exactly two of the roots -1, -w, and \(-w^2\), since the sum of any three of these roots is zero. But this implies that S contains at least one negative root, which contradicts the assumption that the sum of the roots in S is zero, since the negative roots cancel out the positive roots. Therefore, no nonempty proper subset of these roots sums to zero.
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Related Questions
If P(t) is the size of a population at time t, which of the following differential equations describes linear growth in the size of the population?dPdt=200Answer A: d cap p over d t is equal to 200AdPdt=200tAnswer B: d cap p over d t is equal to 200 tBdPdt=100t2Answer C: d cap p over d t is equal to 100 t squaredCdPdt=200PAnswer D: d cap p over d t is equal to 200 cap pDdPdt=100P2
Answers
If P(t) is the size of a population at time t , the differential equation describes linear growth in the size of the population is \(\frac{dP}{dt}=200\)
The differential equation is is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point.
dy/dx = f(x)
Here “x” is an independent variable and “y” is a dependent variable
According to the question,
Size of population at t time = P(t)
Also given ,The differential equations have a linear growth in the size of the population.
So, The degree of the variable must be one. And the equation of the population will be quadratic.
Therefore , dP/dt = 200 tells us the rate change of population with respect to time.
A derivative of linear function is constant .
Hence , option B is correct.
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Suppose a 4x6 coefficient matrix for a system has four pivot columns. Is the system consistent? Why or why not? Choose the correct answer below. O A. There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have seven columns, must have a row of the form [ 0 0 0 0 0 0 1 ], so the system is inconsistent. B. There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have seven columns, could have a row of the form [ 0 0 0 0 0 0 1 ]. so the system could be inconsistent. ] so the system is consistent. OC. There is a pivot position in each row of the coefficient matrix. The augmented matrix will have seven columns and will not have a row of the form [ 0 0 0 0 0 0 1 OD. There is a pivot position in each row of the coefficient matrix. The augmented matrix will have five columns and will not have a row of the form [ 0 0 0 0 1] so the system is consistent.
Answers
The correct answer is (C): There is a pivot position in each row of the coefficient matrix. The augmented matrix will have seven columns and will not have a row of the form [0 0 0 0 0 0 1], so the system is consistent.
If the coefficient matrix has four pivot columns, then it has four leading 1's, one in each row of the matrix. This means that the row-reduced echelon form of the matrix will have four leading 1's and the rest of the entries in those columns will be zero. Since there are no zero rows in the row-reduced echelon form, there cannot be a row of the form [0 0 0 0 0 0 1] in the augmented matrix.
Since there are no zero rows in the row-reduced echelon form, we can conclude that the system of equations is consistent. Furthermore, since there are no free variables (since there are four pivot columns), the system has a unique solution.
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What is the ratio of stars to moons? *
What is the ratio of moons to stars? *
Answers
The ratio of stars to moons and moons to stars are 3:1 and 1:3
What is the ratio?Ratio is described as the comparison of two quantities to determine how many times one obtains the other. The proportion can be expressed as a fraction or as a sign: between two integers.
Given;
There are 3 stars and 1 moon
Now,
Ratio of stars to moon;
=3:1
Ratio of moon to stars;
=1:3
Therefore, the answer ratio will be 3:1 and 1:3
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Bill has $7.30 in dimes and quarters. He has a total of 40 coins. How many quarters and dimes does he have?
Answers
Answer:
35 quarter's and 5 dimes
A box contain 5orange ball, 4yellow and1 greenballa ball i picked out of the box random find the probability that i green, orange, l
Answers
Using the formula for the combination and with the And rule of probability, The answer is 20.
What do you mean by probability?Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes. Statistics is the study of events that follow a probability distribution.
What is combination and it's formula?Combinations are mathematical operations that count the number of potential configurations for a set of elements when the order of the selection is irrelevant. You can choose the components of combos in any order.
It's formula is:
\(C(n,r) = \frac{n!}{r!*(n-r)!}\)
orange balls = 5
yellow balls = 4
green balls = 1
probability = C(5,1)*C(4,1)*C(1,1)
=5*4*1
=20.
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A cylinder has a 10-inch diameter and is 15 inches tallA 6-inch-diameter ball is placed within the cylinderand then the cylinder is filled with waterHow much water is in the cylinder? Give your answer in terms of pi
Answers
Answer:
339n in^3
Step-by-step explanation:
the diameter is the straight line that passes through the centre of a circle and touches the two edges of the circle.
A radius is half of the diameter
Water in the cylinder = volume of cylinder - volume of sphere
volume of a cylinder = nr^2h
n = 22/7
r = radius = 10/2 = 5
n x (5^2) x 15 = 375 in^3
volume of a sphere = (4/3) x (n) x (r^3)
n = 22/7
r = 6/2 = 3
(4/3) x (n) x (r^3) = 36n
In hypothesis testing, if the null hypothesis is rejected, __________. Group of answer choices no conclusions can be drawn from the test the alternative hypothesis must also be rejected the data must have been collected incorrectly the evidence supports the alternative hypothesis
Answers
Therefore, if the null hypothesis is rejected, the evidence supports the alternative hypothesis. This means that the test results provide enough evidence to conclude that the alternative hypothesis is more likely true than the null hypothesis.
In hypothesis testing, if the null hypothesis is rejected, it means that there is sufficient evidence to support the alternative hypothesis. A null hypothesis is a statement that there is no significant difference or relationship between variables, while the alternative hypothesis states that there is a significant difference or relationship. To determine if the null hypothesis is true or not, we conduct a statistical test and calculate the p-value. If the p-value is less than the level of significance, usually set at 0.05, we reject the null hypothesis and accept the alternative hypothesis. Therefore, the correct answer is that the evidence supports the alternative hypothesis. This means that we have found significant results that support our research hypothesis.
Keep in mind that rejecting the null hypothesis does not prove the alternative hypothesis, but it does suggest that it's more plausible based on the data collected.
Therefore, if the null hypothesis is rejected, the evidence supports the alternative hypothesis. This means that the test results provide enough evidence to conclude that the alternative hypothesis is more likely true than the null hypothesis.
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help!
Q. in a supermarket , five chicken cost 18.25 . what is the cost of one chicken ?
Answers
Answer:
each chicken cost 3.65
Step-by-step explanation:
divide 18.25 by 5 and you'll get 3.65
ASAP please!!! Subject: 6th grade Math
A coach needs to fill a rectangular container with water to have available for karate practice. If the dimensions of the container are 30 inches by 22 inches by 20.8 inches, what is the maximum amount of water that the rectangular container will hold?
1,741.6 in3
3,483.2 in3
6,864 in3
13,728 in3
Answers
Answer:
The answer is D, did the test.
Step-by-step explanation:
Answer:
13,728 in.²
Step-by-step explanation:
volume = length × width × height
V = 30 in. × 22 in. × 20.8 in.
V = 13,728 in.²
Use truth tables to determine whether the formula (p∧ ¬q) → (p∧q) is true whenever ¬p is true.
Answers
The formula (p∧ ¬q) → (p∧q) is true whenever ¬p is true. The formula holds true regardless of the truth value of q when p is false.
To determine the truth value of the formula (p∧ ¬q) → (p∧q) when ¬p is true, we need to construct a truth table.
Let's consider the variables p and q and their respective truth values. Since ¬p is true, p must be false.
p q ¬q p∧ ¬q p∧q (p∧ ¬q) → (p∧q)
F T F F F T
F F T F F T
In the truth table above, we evaluate the truth values of the individual components of the formula and then determine the truth value of the entire formula. As we can see, regardless of the value of q, the formula evaluates to true. Thus, the formula (p∧ ¬q) → (p∧q) is true when ¬p is true.
Based on the truth table, we can conclude that the formula (p∧ ¬q) → (p∧q) is true whenever ¬p is true. The formula holds true regardless of the truth value of q when p is false.
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John is 6 years older than frank. 6 years ago, John was twice as old as frank. How old are they both now?
Answers
Answer:
John is 18, Frank is 12.
Step-by-step explanation:
This is th eanswer because if John is older that Frank by 6 years, and ^ years ago, double his age, you have to find a number that is twice the number 6.
12 fits that description:
John Frank
12 6 Let's say these were the ages, twice as old as Frank, but the difference is still 6.
Now, John is 18 (six years later..) and Frank is 12.
I hope you understand the answer and it helps you. :)
9th grade maths solution
Answers
The value of y that satisfies the equation is 3.35 or - 5.35.
What is the value of y?The value of y that satisfies the equation is calculated as follows;
The given equation;
√ (y + 3) + √ ( y - 2) = 5
Square both sides of the equations as follows;
[√ (y + 3) + √ ( y - 2) ]² = 5²
y + 3 + 2(y + 3)(y - 2) + y - 2 = 25
2y + 1 + 2(y² + y - 6) = 25
2y + 1 + 2y² + 2y - 12 = 25
Collect similar terms and simplify the equation;
2y² + 4y - 36 = 0
divide through by 2;
y² + 2y - 18 = 0
Solve the quadratic equation using formula method as follows;
a = 1, b = 2, c = -18
y = 3.35 or - 5.35
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Clare volunteers at a local library during the summer. Her work includes putting labels on 750 books.
How many minutes will she need to finish labeling all books if she takes no breaks and labels 10 books a minute
Answers
Answer:
75 minutes
Step-by-step explanation: 750 divided by 10 is 75
7500 minutes or 125 hours..
I need another answer asap! Please and thank you if you answer this and you will get 15 points! So what is the answer to this question? Dax is buying coffee for 5
people in his office. He also leaves a $2.00
tip for the barista.
If his total, with tip, is $18.25,
how much is each cup of coffee, not including the tip?
Enter your answer as a decimal, like this: 42.53
Answers
Answer:
3.25
Step-by-step explanation:
total 18.25 - 2.00 tip = 16.25
16.25/5=3.25
An aircraft on a reconnaissance mission takes off from its home base and flies 550 miles at a bearing of s 46° e to a location in the sea. it then flies 483 miles from the sea at a bearing of s 55° w to another location. finally, the aircraft flies straight back from the second location to its base. what is the total distance it flies, rounded to the nearest mile? 2 vertical lines. top, point a (airport), point c below are marked on left line. point b marked on middle of right line. abc makes triangle. ab as c equals 5.5. bc as a equals 4.83. interior angle a 46 degrees. exterior angle b 55 degrees. a. 550 miles b. 1,033 miles c. 1,583 miles d. 1,692 miles e. 2,210 miles
Answers
The total distance it flies, rounded to the nearest mile is: d. 1,692 miles .
Total distanceFirst distance= 550 miles (given)
Second distance=483 miles (given)
Now let find the third distance
Let x represent the third distance
Hence:
x/sin79 = 550/sin55
x= 659.1
Now let calculate the total distance
Total distance=550+483+659.1
Total distance= 1692.1
Total distance= 1692 miles (rounded)
Therefore the correct option is d.
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solve for x and find the value of x
Answers
We can write an equation for the interior angles, and we will see that x = 13
How to find the value of x?We want to find the value of x, we can see two adjacent angles on a rectangle, then the sum of these two is equal to 180°, we can write:
3x + 78 + x + 50 = 180
4x + 128 = 180
4x = 180 - 128
4x = 52
x = 52/4 = 13
The value of x is 13.
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Which of the following is the solution set of the
problem?
O (-∞, -3)
(-∞, -3]
O
[-3,00)
O (-3,00)
DONE
Answers
The solution set of the example inequality, 2•x + 3 ≤ -3, is the option;
(-∞, -3]How can the solution set of an inequality be found?A possible inequality that can be used to get one of the options, (the inequality is not included in the question) is as follows;
2•x + 3 ≤ -3Solving the above inequality, we have;
2•x + 3 ≤ -3
2•x ≤ -3 - 3 = -6
2•x ≤ -6
Therefore;
x ≤ -6 ÷ 2 = -3
x ≤ -3
Which gives;
-∞ < x ≤ -3-∞ < x ≤ -3 in interval notation is (-∞, -3]
The solution set of the inequality, 2•x + 3 ≤ -3, is therefore the option;
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Given the function with f(3)=127 and f(1)=95, determine the rate of change over the interval 1≤x≤3
a.16
b.116
c.132
d.32
Answers
Answer:
The answer is 16 :) hope it helps even tho its late
Step-by-step explanation:
Answer:
16
Step-by-step explanation:
jamie is making chocolate cupcakes. the recipe calls for 2/3 cup of cocoa. she wants to make 1/2 of the recipe. how much cocoa will she need?
Answers
Answer:
I think its 1/3
Step-by-step explanation:
one angstrom, symbolized å, is 10-10 m. 1 cm3 = ________ å3.
Answers
one angstrom, symbolized as å, is 10-10 m, so
1 cm^3 = 10^24 angstrom^3
The angstrom, sometimes known as a ångström, is a metric length unit equal to 10^(-10)m, or one ten billionth of a metre, one hundred millionth of a centimeter, 0.1 nanometer, or one hundred picometres. The Swedish alphabet letter serves as its symbol. The Swedish physicist Anders Jonas Angström is commemorated by the unit's name.
The length of an object is measured in centimeters, which are metric units of measurement. The abbreviation is cm.
100 centimeters make up a meter.
One centimeter is equal to ten millimeters.
so one angstrom is 10^-10 m, or 10^-8 cm, so there are 10^8 angstroms in 1 cm.
1 cm^3 = 1 cm 1 cm * 1 cm
1 cm^3 = 10^8 10^8 10^8 angstrom^3
1 cm^3 = 10^24 angstrom^3
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Which segment's length in the geometric mean between RS and ST?
Answers
Answer:
SQ
Explanation:
The measure of the altitude(SQ) drawn from the vertex(Q) of the right angle to the hypotenuse (at point S) is the geometric mean between the measures of the two segments (RS and ST) of the hypotenuse.
Segment SQ is the required segment length.
Read the excerpt from "do not go gentle into that good night." do not go gentle into that good night, old age should burn and rave at close of day; rage, rage against the dying of the light. though wise men at their end know dark is right, because their words had forked no lightning they do not go gentle into that good night. the excerpt's rhyme scheme is
Answers
The rhyme scheme of the excerpt from "Do Not Go Gentle Into That Good Night" is not easily defined by a specific pattern. It combines elements of both regularity and irregularity in its structure. While there is a discernible pattern in some lines, others deviate from that pattern, creating a dynamic and varied effect throughout the poem.
The excerpt exhibits a mix of rhyme patterns, incorporating both end rhyme and internal rhyme. In the first stanza, the rhyme scheme follows an A-B-A-A pattern, with the end words of the first and second lines rhyming ("night" and "light"), while the end words of the third and fourth lines also rhyme ("right" and "night").
However, in the second stanza, the rhyme scheme becomes less structured, with internal rhymes appearing sporadically within lines. For example, "wise men" and "end" have an internal rhyme, as do "dark" and "forked." These occasional internal rhymes contribute to the lyrical and musical quality of the poem, enhancing its emotional impact.
Overall, the rhyme scheme of the excerpt showcases the poet's skillful use of rhyme to convey both the intensity and complexity of the themes explored in the poem.
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a wallet contains six $10 bills, three $5 bills, and six $1 bills (nothing larger). if the bills are selected one by one in random order, what is the probability that at least two bills must be selected to obtain a first $10 bill? (round your answer to three decimal places.)
Answers
Rounded to three decimal places, the probability is approximately 0.457.
To find the probability that at least two bills must be selected to obtain a first $10 bill, we need to consider two scenarios.
Scenario 1: The first bill selected is a $10 bill. There are six $10 bills in the wallet, so the probability of selecting a $10 bill first is 6/15.
Scenario 2: The first bill selected is not a $10 bill. In this case, we need to calculate the probability of not selecting a $10 bill on the first draw, and then selecting a $10 bill on the second draw.
There are nine non-$10 bills in the wallet initially (3 $5 bills + 6 $1 bills), and there are a total of 14 bills left in the wallet after the first draw (15 bills - 1 non-$10 bill).
Therefore, the probability of not selecting a $10 bill on the first draw is 9/15, and the probability of selecting a $10 bill on the second draw is 6/14.
Multiplying these probabilities together, we get (9/15) * (6/14) = 54/210.
To obtain the final probability, we sum the probabilities from both scenarios: (6/15) + (54/210) = 16/35.
Rounded to three decimal places, the probability is approximately 0.457.
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Given g(x) = -x + 5, find g(-6).
Answers
Answer:
g(- 6) = 11
Step-by-step explanation:
substitute x = - 6 into g(x) , that is
g(- 6) = - (- 6) + 5 = 6 + 5 = 11
Answer:
11
Step-by-step explanation:
g(x) = -x + 5
Let x = -6
g(-6) = -(-6) + 5
=6+5
=11
Convert the postfix expression 3 8 7 7 + - 9 3 + to an equivalent infix expression. Show the evolution of the stack
Answers
The postfix expression "3 8 7 7 + - 9 3 +" to an equivalent infix expression and show the evolution of the stack.
Postfix expression: 3 8 7 7 + - 9 3 +
1. Push the first two operands (3 and 8) onto the stack:
Stack: [3, 8]
2. Next, push operands 7 and 7 onto the stack:
Stack: [3, 8, 7, 7]
3. Now, encounter the first operator '+'. Pop the top two operands (7 and 7), perform the addition, and push the result (14) back onto the stack:
Stack: [3, 8, 14]
Infix: (7+7)
4. Next, encounter the operator '-'. Pop the top two operands (14 and 8), perform the subtraction, and push the result (-6) back onto the stack:
Stack: [3, -6]
Infix: (8-(7+7))
5. Push operand 9 onto the stack:
Stack: [3, -6, 9]
6. Push operand 3 onto the stack:
Stack: [3, -6, 9, 3]
7. Encounter the operator '+'. Pop the top two operands (3 and 9), perform the addition, and push the result (12) back onto the stack:
Stack: [3, -6, 12]
Infix: (9+3)
8. Finally, encounter the last operator '-'. Pop the top two operands (12 and -6), perform the subtraction, and push the result (18) back onto the stack:
Stack: [3, 18]
Infix: ((8-(7+7))-(9+3))
The equivalent infix expression is: 3 - ((8 - (7 + 7)) - (9 + 3))
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In the right triangle ABC, where C is the right angle, find all missing parts if B = 43° and a = 32.4. D Find the exact value of cos(15°). Find the exact value of cos(285°).
Answers
The exact value of cos(285°) is (√6 - √2) / 4.
In the right triangle ABC, where C is the right angle, we are given that B = 43° and a = 32.4. We can use trigonometry to find the missing parts.
First, we can use the fact that the angles in a triangle add up to 180° to find angle C:
C = 180° - 90° - 43° = 47°
Now we can use the sine function to find b:
sin(B) = b / c
sin(43°) = b / 32.4
b ≈ 23.9
Finally, we can use the Pythagorean theorem to find the length of side c (the hypotenuse):
c^2 = a^2 + b^2
c^2 = (32.4)^2 + (23.9)^2
c ≈ 40.6
Therefore, the missing parts of the triangle are b ≈ 23.9 and c ≈ 40.6.
To find the exact value of cos(15°), we can use the half-angle formula for cosine:
cos(2θ) = 2cos^2(θ) - 1
If we let θ = 15°, then we have:
cos(30°) = 2cos^2(15°) - 1
cos(15°) = (√3 + 1) / 2√2
Therefore, the exact value of cos(15°) is (√3 + 1) / 2√2.
To find the exact value of cos(285°), we can use the fact that cosine has a period of 360°:
cos(285°) = cos(285° - 360°)
cos(285° - 360°) = cos(-75°)
Since cosine is an even function, we have:
cos(-75°) = cos(75°)
Using the fact that 75° = 45° + 30° and the sum formula for cosine, we have:
cos(75°) = cos(45° + 30°) = cos(45°)cos(30°) - sin(45°)sin(30°)
Since cos(45°) = sin(45°) = √2 / 2 and cos(30°) = √3 / 2 and sin(30°) = 1 / 2, we have:
cos(75°) = (√2 / 2)(√3 / 2) - (√2 / 2)(1 / 2) = (√6 - √2) / 4
Therefore, the exact value of cos(285°) is (√6 - √2) / 4.
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An airplane was cruising at 23,000 feet altitude. Due to turbulence it dropped 4,300 feet.Once the air currents leveled off he went back up 3,600 feet. How far was he from his original cruising altitude
Answers
Answer:
700 ft.
Step-by-step explanation:
need help plz in this question
Answers
Step-by-step explanation:
please mark as brainliest
Help ASAP! 100 Points!! Look AT PHOTO!
Answers
Answer:
1
2,3,4
Step-by-step explanation: to
when the number of football is 1 is less expensive than basketball
Using suitable identity, find the value of 87^3+ 13^3/
87^2 −87 ×13 + 13^2
Answers
The value of the given expression [\(87^3+ 13^3/87^2 -87 * 13 + 13^2\)] by simplifying the numerator and denominator using suitable identities is 100.
We will first calculate the numerator:
As (\(a^3\) + \(b^3\)) = (a + b)(\(a^2\) - ab + \(b^2\)) :
\(87^3\) + \(13^3\) = (87 + 13)(\(87^2\) - \(87 * 13\) + \(13^2\))
= 100(\(87^2\) - 87 * 13 + \(13^2\))
Now, calculate the denominator:
\(87^2 - 87 * 13 + 13^2\)
As,(\(a^2 -2ab +b^2\)) =\((a - b)^2\):
\(87^2 - 87 * 13 + 13^2 = (87 - 13)^2\)
\(= 74^2\)
So by solving the equation further:
\((87^3+13^3) / (87^2- 87 * 13+13^2) = 100*(87^2- 87 *13 + 13^2)/(87^2 - 87 * 13 + 13^2)\)
As we can see the numerator and denominator are the same expressions (\(87^2 - 87 * 13 + 13^2\)). so, they cancel each other:
\((87^3 + 13^3) / (87^2 - 87 * 13 + 13^2) = 100\)
So, the value of the given expression is 100.
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