Mean value of the function = E(X) = ∞/πOption (b) ∞/π is the correct option.
Given a density function of a random variable X as f(x)={ 2 π , 0, 0. The question is to find the mean value of the function.As we know,The mean value of the function = E(X) = ∫xf(x)dx, where x is the random variable and f(x) is the density function,∫ denotes integral from negative infinity to infinityOn substituting the given values,∫xf(x)dx= ∫x (2/π)dx= (2/π) ∫xdx= (2/π)(x^2/2)+C ……(1)Where C is the constant of integration.But given the density function is zero for all negative x values and f(0) = 2/π, so the integral should be calculated from 0 to infinity instead of negative infinity to infinity.On substituting the values,∫0∞ x (2/π)dx= (2/π) ∫0∞xdx= (2/π) (x^2/2) [0,∞]= ∞/πTherefore, mean value of the function = E(X) = ∞/πOption (b) ∞/π is the correct option.
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Which BEST describes 2
perpendicular lines?
a) They are the same
b) They are negative reciprocals of each other
c) They are the same line
d) They have the same slopes
e) They have slopes that are negative reciprocals of each other.
The BEST choice for two perpendicular lines is: e) Their slopes are the reciprocal negatives of one another.
How could you tell if two lines were perpendicular to one another?Perpendicular lines are those that cross at a correct angle when two separate lines share a plane. Vertical and horizontal lines, or the axes of a coordinate plane, are perpendicular to one another. Two parallel lines' slopes have negative reciprocal slopes.
If two lines intersect at the a right angle in Euclidean geometry, they are said to be parallel (90 degrees).
When two lines are parallel, their slopes are the reciprocal of each other's negative values. A slope of the second line would be "-1/m" if the curve of the initial line is "m."
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Point D is the centroid of ∆ABC, DC = 8x - 6, and ED = 3x + 2. What is CE?
Answer:
See below.
Step-by-step explanation:
For this problem, we are asked to solve for CE.
Because Point D is a Centroid, CE is a median.
What is a Median?
A Median is a line that connects from 1 vertex of a Triangle, through the Centroid, and then ending on the midpoint of the opposite side.
The distance from the Triangles' Vertex to the Centroid is 2 times the distance from the Centroid to the Midpoint.
Basically;
[tex]ED = \frac{1}{2} CD[/tex]
Let's solve for x first.
[tex]3x+2=\frac{1}{2} (8x-6)[/tex]
Distribute:
[tex]3x+2=4x-3[/tex]
Subtract 4x from both sides:
[tex]-x+2=-3[/tex]
Subtract 2 from both sides:
[tex]-x=-5[/tex]
Divide by -1:
[tex]x=5.[/tex]
CD, and DE are 2 parts of CE. When we add CD, and DE together we will have the value of CE.
[tex]CD+DE=CE.[/tex]
Let's identify CD and DE first since we have the value of x.
[tex]CD=8(5)-6=34.[/tex]
[tex]DE=3(5)+2=17.[/tex]
Add:
[tex]34+17=51 \ (CE)[/tex]
Our final answer is D, CE = 51.
Find an equation of the line with gradient 1/3 and passes through the point (5,-1)
Answer:
y = x/3 - 8/3 (OR) 3y = x - 8
Both of the above solutions are the same
Step-by-step explanation:
Using the form 'y=mx+c',
Since m = 1/3,
y = 1/3x + c
Substituting (5, -1) into the above equation:
-1 = 5/3 + c
c = -1 - 5/3
= -8/3
Hence,
y = x/3 - 8/3
(which is also the same as)
3y = x - 8
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A number ,x, rounded to 2 decimal places is 15. 38 whats the error interval for x
To calculate the error interval for x, we must analyse the probable range of values for x before it was rounded to 15.38. Because x was rounded to two decimal places, we know it is between two values.
The biggest number that can be rounded down to 15.38 is 15.375, and the smallest number that can be rounded up to 15.38 is 15.385. As a result, the error range for x is [15.375, 15.385]. The error interval for a rounded integer is the range of possible values for the original number before rounding. The integer x was rounded to 15.38 with two decimal places in this example. To calculate the error interval Consider the highest and lowest feasible numbers for x that might have been rounded to 15.38. We consider the number that ends in.385 for the highest possible value. If x was larger or equal to this value, it was rounded up to 15.39. As a result, the greatest possible value for x is 15.385. Consider the number that ends in.375 for the least feasible value. If x was less than or equal to this value, it was rounded down to 15.37. As a result, the least number that x might have been is 15.375. As a result, the error interval for x is comprised of values ranging from 15.375 to 15.385, inclusive. This might be any number.
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Pls help help Algebra1
Answer:
(x-1)(x+6)
Step-by-step explanation:
Consider drawing four cards without replacement from a standard deck of cards. What is the conditional probability of drawing a heart on the fourth draw, given that the first three cards are not hearts?
Group of answer choices:
13/52 or 1/4
10/49
10/52 or 5/26
13/49
The conditional probability of drawing a heart on the fourth draw given that the first three cards are not hearts is 13/49.. The correct answer is D.
After drawing three cards without replacement, there are 52 - 3 = 49 cards remaining in the deck, of which 13 are hearts and 36 are not hearts.
If the first three cards are not hearts, then there are 36 cards remaining in the deck for the fourth draw, of which 13 are hearts and 23 are not hearts.
The conditional probability of drawing a heart on the fourth draw, given that the first three cards are not hearts, can be calculated using Bayes' theorem:
P(heart on 4th draw | first 3 not hearts) = P(heart on 4th draw and first 3 not hearts) / P(first 3 not hearts)
The probability of drawing a heart on the fourth draw and the first three not being hearts is:
P(heart on 4th draw and first 3 not hearts) = (36/52) * (35/51) * (34/50) * (13/49) = 0.0334
The probability of the first three cards not being hearts is:
P(first 3 not hearts) = (36/52) * (35/51) * (34/50) = 0.5764
Therefore, the conditional probability of drawing a heart on the fourth draw, given that the first three cards are not hearts, is:
P(heart on 4th draw | first 3 not hearts) = 0.0334 / 0.5764 ≈ 0.058 or approximately 5.8%.
Therefore, the answer is 13/49. The correct answer is D.
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What percent of 25 Doctors equals 2
The percent of 25 doctors which is equivalent to 2 as required to be determined is; 8%.
What is the equivalent of 2 out of 25 doctors?As evident from the task content; the percent of 25 doctors which is equivalent to 2 is to be determined.
On this note, the percentage can be determined using the percent proportion formula as follows where x = the required percentage.
x / 100 = 2 / 25
25x = 200
x = 200 / 25
x = 8.
Hence, the equivalent percentage as required to be determined is; 8%.
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help asap!!!!!!!!!!!!!!!!
4. (a) circumference of the figure: 2 * 22/7 * 3 = 18.86 cm
(b) Area of the figure: 22/7 * 3² = 28.29 cm²
5. (a) circumference of the figure: 22/7 * 35 = 110 ft
(b) Area of the figure: 22/7 * (35/2)² = 962.5 ft²
How to find the area and circumference of a circle?The area of a circle is given by the formula:
A = πr²
where r is the radius of the circle
The circumference of a circle is given by the formula:
C = 2πr or πd (Recall: d = 2r)
where r is the radius and d is the diameter of the circle
No. 4
r = 3 cm
(a) C = 2πr
C = 2 * 22/7 * 3 = 18.86 cm
(b) A = πr²
A = 22/7 * 3² = 28.29 cm²
No. 5
d = 35 ft
(a) C = πd
C = 22/7 * 35 = 110 ft
(b) A = πr²
A = 22/7 * (35/2)² = 962.5 ft²
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Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at x=−4 and x=2Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).f(x)=
The function f(x)= (x+4)(x-2)/(x+4)(x-2) passes through the origin with a constant slope of 1, and has two removable discontinuities at x=-4 and x=2.
The function that passes through the origin with a constant slope of 1 and has removable discontinuities at x=-4 and x=2 is given by:
f(x)= (x+4)(x-2)/(x+4)(x-2). This function can be written in its factored form as f(x)= 1.
The function has a constant slope of 1, which means that for any two points (x_1,y_1) and (x_2,y_2) the slope is given by
(y_2-y_1)/(x_2-x_1)=1.
The function has two removable discontinuities at x= -4 and x= 2. This means that at these points the function is undefined, and the derivative of the function is infinite.
Removable discontinuities can be resolved by factoring out a common factor from the numerator and denominator of the function. In the case of the given function, factoring out (x+4) and (x-2) resolves the discontinuities. The graph of the function is a straight line which passes through the origin. This is because the function is of the form y=mx, where m is the constant slope of 1, which makes the function pass through the origin.
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unit 7 right triangle and trigonometry quiz 7-2
Yes, trigonometry only works on the right angled triangles.
Explain about the Right triangle and trigonometry?The study of correlations between triangles' side lengths and angles is known as trigonometry.
A triangle with one right angle is referred to as a right triangle as well as right-angled triangle. Trigonometry's foundation is the relationship between a right triangle's sides and angles.The area of mathematics called trigonometry deals with calculating triangles' unknowable sides and angles.There are numerous uses for trigonometry in both engineering and science. We will just provide a couple of instances from surveying as well as navigation in this section.Thus, the right angle, or 90°, is always one angle. The hypotenuse is the side with the 90° angle opposite. The longest side is always the hypotenuse. The other two inner angles add up to 90 degrees.
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The complete question is-
Right triangle and trigonometry:
Does trigonometry work on right triangles?
i need help on this question
Answer:[tex]180\pi[/tex]
Step-by-step explanation:
[tex]v=\pi rx^{2} h\\v=\pi 6^{2} 5\\\\v=180\pi[/tex]
Write the first 5 terms of a sequence whose general term, An, is given as an =-9n-9
The first 5 terms of the sequence, whose general term is an = -9n - 9, would be the following: -18, -27, -36, -45, -54.
How to Calculate the Terms of a Sequence?In mathematics, a sequence is an ordered list of numbers, called terms, that follow a particular pattern or rule. Each term in a sequence is identified by its position in the sequence, which is usually represented by an integer index or subscript.
To find the first 5 terms of the sequence with the general term an = -9n - 9, we can simply substitute the values of n from 1 to 5 into the formula and simplify:
a1 = -9(1) - 9 = -18
a2 = -9(2) - 9 = -27
a3 = -9(3) - 9 = -36
a4 = -9(4) - 9 = -45
a5 = -9(5) - 9 = -54
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Illustrate a probability distribution of a random variable X showing the number of face mask sold per day and its corresponding probabilities.
DAY. X
1. 14
2. 15
3. 10
4. 12
5. 10
6. 15
7. 14
8. 15
9. 20
10. 25
Looking at the corresponding probability in the distribution will allow you to determine the likelihood of selling a specific quantity of face masks each day.
Here is the probability distribution for the number of face masks sold per day:
X P(X)
10 0.2
12 0.1
14 0.2
15 0.3
20 0.1
25 0.1
To find the probability of selling a certain number of face masks per day, you look at the corresponding probability in the distribution. For example, the probability of selling 15 face masks per day is 0.3.
Note that the probabilities sum to 1, which is a requirement for any probability distribution.
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A plane can fly 340 mph in still air. If it can fly 200 miles downwind in the same amount of time it can fly 140 miles upwind, find the velocity of the wind.
By using the speed fοrmula, the velοcity οf the wind is apprοximately 66.23 mph.
What is speed?Speed is defined as the distance travelled by an οbject in a given amοunt οf time. Speed is a scalar quantity, meaning that it has magnitude but nο directiοn. Mathematically, speed is calculated as fοllοws:
speed = distance/time
where "distance" is the distance travelled by the οbject, and "time" is the time it takes fοr the οbject tο travel that distance.
Let's assume that the speed οf the wind is represented by the variable "v" (in mph). When the plane flies dοwnwind, the speed οf the plane relative tο the grοund is the sum οf its airspeed and the wind speed, οr (340 + v) mph. Similarly, when the plane flies upwind, the speed οf the plane relative tο the grοund is the difference between its airspeed and the wind speed, οr (340 - v) mph.
Nοw, lets use the fοrmula:
time = distance / speed
tο set up twο equatiοns and sοlve fοr the wind speed "v".
First, fοr the dοwnwind flight:
time = distance / speed
200 / (340 + v) = t
Secοnd, fοr the upwind flight:
time = distance / speed
140 / (340 - v) = t
Since the time taken fοr bοth flights is the same, we can equate the twο equatiοns:
200 / (340 + v) = 140 / (340 - v)
Nοw, we can sοlve fοr "v":
200(340 - v) = 140(340 + v)
68,000 - 200v = 47,600 + 140v
308v = 20,400
v ≈ 66.23
Therefοre, the velοcity οf the wind is apprοximately 66.23 mph.
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House Loan
Cost: $450,000
Length of Loan: 30 Years
Simple Interest Rate: 6.00%
Yearly Taxes: $2,000
Yearly Insurance: $1,500
What are your Monthly Payments with taxes & insurance:
Step-by-step explanation:
Your monthly payments with taxes and insurance included would be $2,903.71. This is calculated by taking the loan amount of $450,000 and multiplying it by the simple interest rate of 6.00%. The result is $27,000, which is then divided by the length of the loan, 30 years. This gives you
the principal and interest portion of your monthly payment, which is $2,033.33. To that, you add your yearly taxes of $2,000 and insurance of $1,500, divided by 12 months, to get an additional $416.67 and $125, respectively. Adding these two numbers together gives you your total monthly payment of $2,903.71.
A couple has $5,000 to invest and has to choose between three investment options.
• Option A: 2.25% interest applied each quarter
• Option B: 3% interest applied every 4 months
• Option C: 4.5% interest applied twice each year
If they plan on no deposits and no withdrawals for 5 years, which option will give them the largest
balance after 5 years? Use a mathematical model for each option to explain your choice. Share the
value of each investment.
Make sure you use the formula from the lesson for each option then compare. As an investment, you
want the option with the most money!
nt
A = P(1 +r )"nt
—
n
Deshaun runs each lap in 4 minutes. He will run at most 7 laps today. What are the possible numbers of minutes he will run today?
Use t for the number of minutes he will run today.
Write your answer as an inequality solved for t.
this question is an inequality o please write ur answer as one
The inequality is t ≤ 28.
What is inequality?Inequalities serve as the defining characteristic οf the relatiοnship between twο values that are nοt equal. Equal dοes nοt imply inequality. Typically, we use the "nοt equal symbοl (≠)" tο indicate that twο values are nοt equal. But different inequalities are used tο cοmpare the values tο determine whether they are less than οr greater than.
Given that Deshaun runs each lap in 4 minutes. He will run at mοst 7 laps tοday.
Assume that t fοr the number οf minutes he will run tοday.
He will run at mοst 7×4 = 28 minutes.
The symbοl οf at mοst is ≤.
The inequality is t ≤ 28
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The volume of a triangular pyramid is 273 units³. If the base and height of the
triangle that forms its base are 14 units and 9 units respectively, find the height of the
pyramid.
the height of the triangular pyramid is 13 units.
How to solve?
The formula for the volume of a pyramid is given by V = (1/3)Bh, where B is the area of the base and h is the height of the pyramid. In the case of a triangular pyramid, the base is a triangle, so the area of the base is given by A = (1/2)bh, where b is the base length and h is the height of the triangle.
Given that the volume of the triangular pyramid is 273 units³, we have:
V = (1/3)Bh = 273
We know that the base of the triangular pyramid has a base length of 14 units and a height of 9 units, so its area is:
A = (1/2)bh = (1/2)(14)(9) = 63
Substituting this value for B in the equation for the volume, we have:
(1/3)(63)h = 273
Multiplying both sides by 3, we get:
63h = 819
Dividing both sides by 63, we get:
h = 13
Therefore, the height of the triangular pyramid is 13 units.
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The Texas Lottery lists the following probability for winning with a ticket.
Winnings Probability $300,000 1/5,518,121 $70,000 1/1,715,449 $600 1/9,024 $18 1/1,768 $5 1/358 $3 1/166 (a) What is the probability you win a positive amount of money? Round to two decimal places (b) What would the ticket have to cost (in positive dollars) for this lottery to be fair, in the sense that your expected profit is $
A) the probability of winning a positive amount of money is approximately 0.000487.
B) the ticket would have to cost $0.0522 (or approximately $0.05) for this lottery to be fair, in the sense that the expected profit is $0.
What is the justification for the above response?
(a) To calculate the probability of winning a positive amount of money, we need to add up the probabilities of winning each prize except for the $0 prize:
Probability of winning a positive amount
= Probability of winning $300,000 + Probability of winning $70,000 + Probability of winning $600 + Probability of winning $18 + Probability of winning $5 + Probability of winning $3
Probability of winning a positive amount
= 1/5,518,121 + 1/1,715,449 + 1/9,024 + 1/1,768 + 1/358 + 1/166
Probability of winning a positive amount
≈ 0.000487 (to two decimal places)
Thus, the probability of winning a positive amount of money is approximately 0.000487.
(b) To find the ticket cost that makes the lottery fair, we need to calculate the expected profit, which is the sum of the products of the probability of winning each prize and the amount of money won for that prize. Since there are six possible prizes, the expected profit can be written as:
Expected profit = (1/5,518,121)($300,000) + (1/1,715,449)($70,000) + (1/9,024)($600) + (1/1,768)($18) + (1/358)($5) + (1/166)($3) - Ticket cost
Simplifying the expression above, we get:
Expected profit
= $0.0522 - Ticket cost
For the lottery to be fair, the expected profit should be $0. Therefore, we can set the expected profit equal to $0 and solve for the ticket cost:
$0 = $0.0522 - Ticket cost
Ticket cost = $0.0522
Therefore, the ticket would have to cost $0.0522 (or approximately $0.05) for this lottery to be fair, in the sense that the expected profit is $0.
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Rajan brought a book for Rs 180 and sold it to sajan at a profit of 20%. Sajan sold that book to Nirajan at a loss of20%. At what price Nirajan should sell the book to receive 5% profit.
Answer:
Ans: Rs 181.44.
Step-by-step explanation:
what do we need to add to 2 / 1 8 to make 5
Answer:
[tex]4\frac{8}9}[/tex]
Step-by-step explanation:
We can do this mathematically.
[tex]5-\frac{2}{18} = \frac{90}{18} - \frac{2}{18} = \frac{88}{18} = \frac{44}{9} = 4\frac{8}{9} \\[/tex]
We can also work this logically.
We know that [tex]\frac{2}{18}[/tex] is less than 1, so our answer should be (5-1) and a fraction.
Our answer should be 4 plus a fraction less than 1. To find what we need to add to [tex]\frac{2}{18}[/tex] to make 1, we ask ourself how many eighteenths make 1. So if we need to 18 parts and we have 2 already, we add 16 eighteens to make 1. The answer is [tex]4 \frac{16}{18}[/tex], or simplified, [tex]4\frac{8}9}[/tex].
Find the sum and product of the complex numbers \( 2-3 i \) and \( -2+6 i \). The sum is (Type your answer in the form \( a+b i \).) The product is (Type your answer in the form \( \mathrm{a}+\mathrm{
The given complex numbers are 2 - 3i and -2 + 6i. We need to find the sum and product of the given complex numbers.
Sum of the complex numbers = (2 - 3i) + (-2 + 6i)
= 2 - 2 + (-3i + 6i)
= 4i
Therefore, the sum of the given complex numbers is 4i.
Now, let's find the product of the given complex numbers.
Product of the complex numbers = (2 - 3i) (-2 + 6i)
= -4 + 12i + 6i - 18i^2
= -4 + 18i + 18 (as i^2 = -1)
= 14 + 18i
Therefore, the product of the given complex numbers is 14 + 18i.
Hence, the sum of the given complex numbers is 4i and the product of the given complex numbers is 14 + 18i.
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Marta solved an equation. Her work is shown below. equation: 2(x-4) + 2x = x+7 line 1: 2x 81+ 2x = x + 7 line 2: 4x8=x+7 line 3: 3x - 8 = 7 line 4: 3x = 15 line 5: x = 5 Which step in Marta's work is justified by the distributive property?
Step-by-step explanation:
The distributive property of multiplication over addition states that for any numbers a, b, and c, a × (b + c) = a × b + a × c.
Looking at Marta's work, we can see that the distributive property was used in line 1, where she distributed the 2 to both terms inside the parentheses:
2(x - 4) + 2x = x + 7
2x - 8 + 2x = x + 7
4x - 8 = x + 7
Therefore, the step in Marta's work that is justified by the distributive property is line 1, where she distributed the 2 to both terms inside the parentheses.
In the scale drawing of the base of a rectanglar swimming pool, the pool is 8. 5 inches long and 4. 5 inches wide. If 1 inch on the scale drawing is equivalent to 4 meters of actual length, what are the actual length and width of the swimming pool
The actual length and width of the swimming pool are 34 meters and 18 meters, respectively, based on a scale of 1 inch to 4 meters.
Based on the given information, we know that the scale of the drawing is 1 inch to 4 meters. This means that every 1 inch on the drawing represents 4 meters in actual length.
The length of the swimming pool on the drawing is given as 8.5 inches. To find the actual length of the swimming pool, we need to multiply the length on the drawing by the scale factor:
Actual length of the swimming pool = 8.5 inches x 4 meters/inch = 34 meters
Similarly, the width of the swimming pool on the drawing is given as 4.5 inches. To find the actual width of the swimming pool, we can use the same formula:
Actual width of the swimming pool = 4.5 inches x 4 meters/inch = 18 meters
Therefore, the actual length of the swimming pool is 34 meters and the actual width is 18 meters.
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The low temperature on Monday was 6°F greater than half the temperature on Sunday. If the low temperature on Monday was 5°F, what was the low temperature on Sunday in degrees Fahrenheit?
Answer:
-2°F
Step-by-step explanation:
Let M and S represent Monday and Sunday's low temperatures respectively. The given information relates M and S with the equation:
[tex]M = 6 + \frac{S}{2}[/tex]
We can rearrange the equation to solve for S.
[tex]S = 2(M-6)[/tex]
We can substitute 5 for M and find that S = -2.
∴The low temperature on Sunday was -2°F.
.
I need some help with it
The Answer is approximately equal to 942.48 or 300π
(π=pi)
Tree house Base, Scale of 1 : 21:
Model: ? in. Actual: 7 ft
The length of the tree house with a base of 7 ft on the model is given as follows:
1/3 ft.
How to obtain the length of the model?The length of the model is obtained applying the proportions in the context of the problem.
The scale is a ratio that relates the size of a drawing or map to the actual size of the object or area being represented. It indicates how much the drawing or map has been reduced or enlarged in size from the actual object or area.
Hence the scale of 1:21 means that each ft on the drawing represents 21 ft of actual distance, hence the length of the drawing for an actual distance of 7 ft is given as follows:
1/21 x 7 = 1/3 ft.
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Salim walked 1 1/3 miles to school and then 2 1/2
miles to work after school. How far did he walk in all?
In response to the query, we can state that Salim thereby covered a total fraction distance of about 3.83 miles.
what is fraction?To represent a whole, any number of equal parts or fractions can be utilised. In standard English, fractions show how many units there are of a particular size. 8, 3/4. Fractions are part of a whole. In mathematics, numbers are stated as a ratio of the numerator to the denominator. They can all be expressed as simple fractions as integers. In the numerator or denominator of a complex fraction is a fraction. The numerators of true fractions are smaller than the denominators. A sum that is a fraction of a total is called a fraction. You can analyse something by dissecting it into smaller pieces. For instance, the number 12 is used to symbolise half of a whole number or object.
Salim therefore walked a total of:
4/3 Plus 5/2 miles
We must identify a common denominator in order to add these fractions:
Six is the least frequent multiple of 3 and 2.
We may therefore rephrase the following fractions with denominators of 6:
4/3 = (4/3) x (2/2) = 8/6
5/2 = (5/2) x (3/3) = 15/6
We can now combine the fractions:
8/6 + 15/6 = 23/6
Salim covered a distance of 23/6 miles in all.
This fraction can be simplified by dividing both the numerator and denominator by their largest common factor, which is 1:
3.83 miles, or 23/6 miles (rounded to two decimal places)
Salim thereby covered a total distance of about 3.83 miles.
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suppose that the matrix has repeated eigenvalue with the following eigenvector and generalized eigenvector: with eigenvector and generalized eigenvector write the solution to the linear system in the following forms.
The solution to the linear system with repeated eigenvalues can be written in two forms: the exponential form and the Jordan form. In the Jordan form, the solution is written as: [tex]x(t) = e^(J*t) * v_0[/tex]
In the exponential form, the solution is written as:
[tex]x(t) = e^(lambda*t) * (v + t*w)[/tex]
Where lambda is the repeated eigenvalue, v is the eigenvector, and w is the generalized eigenvector.
In the Jordan form, the solution is written as : [tex]x(t) = e^(J*t) * v_0[/tex]
Where J is the Jordan matrix, which is a matrix with the repeated eigenvalue lambda on the diagonal and a 1 on the superdiagonal, and v_0 is the initial condition vector.
Both forms of the solution give the same result, but the exponential form is simpler and easier to compute. The Jordan form is more general and can be used to find the solution for any linear system with repeated eigenvalues.
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let be an integral domain with a descending chain of ideals . suppose that there exists an such that for all . a ring satisfying this condition is said to satisfy the descending chain condition, or dcc. rings satisfying the dcc are called artinian rings, after emil artin. show that if satisfies the descending chain condition, it must satisfy the ascending chain condition.
As before, it follows that A3 is a maximal ideal of R, contradicting the fact that the chain is infinite. Therefore, R satisfies the ACC.
It is given that be an integral domain with a descending chain of ideals. Suppose that there exists an n such that for all i ≥ n, then ai = an. A ring satisfying this condition is said to satisfy the descending chain condition or DCC. Rings satisfying the DCC are called Artinian rings, after Emil Artin.
The statement to be proved is if R satisfies the descending chain condition, it must satisfy the ascending chain condition. Suppose, by contradiction, that R satisfies the DCC but does not satisfy the ACC. Then, there is an infinite ascending chain: A1 ⊂ A2 ⊂ A3 ⊂ A4 ⊂ ···.
Note that if R is an integral domain and if a ∈ R, then (a) is either (0) or is a maximal ideal in R. Hence, (0) is a minimal element in the collection of all proper ideals of R. Suppose A1 is a proper ideal of R that is maximal with respect to not being finitely generated. Since R satisfies the DCC, A1 cannot be infinite. Therefore, A1 is a finite set. Suppose A1 is not principal.
Then there exist two elements a, b ∈ A1 that do not belong to (a) and (b) respectively. This means that (a, b) is a proper ideal of R, properly containing A1, which contradicts the maximality of A1. Thus, A1 is a principal ideal generated by an element a1 ∈ A1.Suppose A2 = (a1, a2, a3, · · · , am) is a proper ideal properly containing A1. If A2 is finitely generated, then A2 ⊃ (a1) ⊃ (0) is a finite descending chain of ideals, which contradicts the DCC.
Thus, A2 is not finitely generated. By the maximality of A1, A2 must be principal, generated by an element a2 ∈ A2. It follows that a2 = c1a1 + c2a2 + · · · + cmam, where ci ∈ R for all i. Hence, (1 − c2)a2 = c1a1 + · · · + cmam, which means that a2 ∈ (a1). Therefore, (a1) = A1 = A2, and it follows that A2 is a maximal ideal of R.
Suppose A3 is a proper ideal properly containing A2. If A3 is finitely generated, then A3 ⊃ A2 ⊃ (0) is a finite ascending chain of ideals, which contradicts the ACC. Thus, A3 is not finitely generated. By the maximality of A2, A3 must be principal, generated by an element a3 ∈ A3.
As before, it follows that A3 is a maximal ideal of R, contradicting the fact that the chain is infinite. Therefore, R satisfies the ACC.
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