The matrix norm ||B||_M is equal to the maximum value of ||Bx||_M over all vectors x with a Euclidean norm of 1, i.e., ||B||M = max{||x||₂ = 1} ||Bx||_M.
To show that the matrix norm ||B||M = max{||x||₂ = 1} ||Bx||₂, we need to demonstrate two properties
the upper bound property and the achievability property.
Upper bound property:
We want to show that ||B||M ≤ max{||x||₂ = 1} ||Bx||₂.
Let's consider an arbitrary vector x with ||x||₂ = 1. Since ||Bx||₂ represents the Euclidean norm of the vector Bx, it follows that ||Bx||₂ ≤ ||Bx||_M for any x. Therefore, taking the supremum over all such x, we have:
sup{||Bx||₂ : ||x||₂ = 1} ≤ sup{||Bx||_M : ||x||₂ = 1}.
This implies that
||B||M ≤ max{||x||₂ = 1} ||Bx||_M.
Achievability property:
We want to show that there exists a vector x such that ||x||₂ = 1 and
||Bx||M = max{||x||₂ = 1} ||Bx||_M.
Consider the vector x' that achieves the maximum value in the expression max_{||x||₂ = 1} ||Bx||_M. Since the maximum value is attained, ||Bx'||M = max{||x||₂ = 1} ||Bx||_M.
Since ||x'||_2 = 1, we have ||Bx'||₂ ≤ ||Bx'||_M. Therefore,
||Bx'||₂ ≤ ||B'||M = max{||x||₂ = 1} ||Bx||_M.
Combining both properties, we conclude that
||B||M = max{||x||₂ = 1} ||Bx||_M.
In summary, we have shown that the matrix norm ||B||_M is equal to the maximum value of ||Bx||_M over all vectors x with a Euclidean norm of 1, i.e., ||B||M = max{||x||₂ = 1} ||Bx||_M.
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Part A
The prefix inter: means "between," and the word part cept means "to take." Based on
this information, what does the word interceptor mean as it is used in the passage?
"The interceptors covertly eavesdropped on Germany's radio messages, sending
them along to the team at Bletchley." (paragraph 6)
one who broadcasts radio messages to bring down his enemy
one who cannot decide between two sides, as in a war or argument
one who seizes something on its way from one place to another
one who takes part in cracking codes by means of a hidden key
Answer: B - one who cannot decide between two sides, as in war or argument
Step-by-step explanation:
There is a cohort study assessing the potential association between exercise and future cardiac events. A risk ratio less than one suggests: There does not appear to be an association between exercise and future cardiac events. There appears to be a positive association between exercise and future cardiac events. There appears to be a negative association between exercise and future cardiac events. None of the answers listed here are correct.
A risk ratio less than one suggests a negative association between exercise and future cardiac events.
In a cohort study, the risk ratio (also known as relative risk) is a measure used to assess the association between an exposure (in this case, exercise) and an outcome (future cardiac events). The risk ratio compares the risk of experiencing the outcome in the exposed group (those who exercise) to the risk in the unexposed group (those who do not exercise).
When the risk ratio is less than one, it indicates a lower risk of the outcome among the exposed group compared to the unexposed group. In this context, a risk ratio less than one suggests a negative association between exercise and future cardiac events. It implies that individuals who engage in regular exercise have a reduced risk of experiencing cardiac events compared to those who do not exercise.
It is important to note that a risk ratio less than one does not imply a complete absence of risk or a guarantee of protection. It simply indicates a lower relative risk among individuals who exercise. Other factors and confounding variables should also be considered in interpreting the results of a cohort study and drawing conclusions about the association between exercise and future cardiac events.
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9. The lengths of the sides of a triangle are 8 and 12. The length of the third side must be between what
two numbers? Repeat with other side lengths and write a generalization for sides of lengths a and b,
where b>a.
Answer:
b>a
Step-by-step explanation:
correct just divide then subtract
Charlotte wants to paint a room that measures 15 ft wide, 20 ft long, and 8 ft high. One gallon of the paint she selects covers 400 ft² and costs $16.99. The sales tax is 6%. She plans to paint the ceiling and the four walls with the same paint. She ignores the area of the two windows and the door into the room when she estimates the amount of paint she needs. How many gallons of paint should she buy? How much will it cost?
The amount of paint she should buy is 2 gallons and it will cost $36.02.
What is area of shape?The area of a shape is the space occupied by the boundary of a plane figures like circles, rectangles, and triangles.
The room takes the shape of a cuboid with the dimensions 30 by 15 by 8
Charlotte will be left with 5 surfaces to paint i.e
The ceiling and the other 4 walls.
Therefore the total surface area of the portion to be painted = lh + lh + lb + bh + bh
where l = 20, b = 15 , h = 8
= 20 × 8 +120× 8 + 20×15 + 15× 8 + 15 × 8
= 160+160+300+120+120
= 860 ft²
Let's assume the windows and doors take area of 60ft²
The area to be painted is 800ft²
1 gallon of paint will go for 400ft²
therefore the number of gallon for 800ft² = 800/400 = 2 gallons.
The cost of 1 gallon is $ 16.99
2 gallons = 2 × 16.99 = $33.98
Sales tax = 6%
= 6/100 × 33.98
= $2.04
Total cost = 33.98 + 2.04
= $36.02
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Pls help me I need some help.
Answer:
scale factor is 3
Step-by-step explanation:
Answer:
3
Step-by-step explanation:
5:15. 8:24. 9:27. they are all a ratio of 3
In the pic below help my cousin out UnU I don’t wanna help em ;-;
Answer:
The mode is 78, for it keeps repeating.
There is a manhole cover with a radius of 1 foot. What is the manhole cover's diameter?
Step-by-step explanation:
This seems like you just want to figure out the circumference of the manhole cover. The formula for the circumference of a circle is pi (3.14) multiplied by the diameter (d) of the circle so, circumference=πd. (π is the symbol for pi and approx. equals 3.14)
Circumference = πd
= 3.14(d)
= 3.14(3)
= 9.42 ft.
The length of the brass grip-strip will be 9.42 ft.
If the problem was stated in terms of the radius of the manhole cover then the formula would be circumference = 2πr which is the radius multiplied by 2 then multiplied by pi.
The radius of a circle is the distance from the center to the edge and the diameter is the distance from one edge of the circle to the other passing through the center of the circle.
Well, if the grip strip were of no width and could be straightened out to a line (which a piece of rubber cut in a circle couldn't be), then the length of the grip would correspond to the circumference of the manhole cover.
Circumference = 2*PI*radius = PI*diameter so your answer is 3*PI feet long.
For the following regression model Y = α + βX + u
- Specify the assumptions that must hold, so the least-squares estimators of α and β are BLUE.
The ordinary least-squares estimators are used in regression analysis. In order to make the least-squares estimators of α and β BLUE, some assumptions must be met.
Assumptions of Linear Regression Model: In order to apply the OLS estimation technique, the following assumptions must be met :
Linearity - The linear regression analysis's relationship between independent and dependent variables should be linear.
If this assumption is not met, the linear regression will not be able to offer a good estimate of the relationship between variables.
Homoscedasticity - The variances of the residuals should be equal for all values of X.
In addition, the error term should have a constant variance. When this assumption is not met, the model is referred to as heteroscedasticity. Normality - The error term must be normally distributed.
This assumption's significance is critical since OLS depends on normality, and the model's failure to meet this assumption can be disastrous.
Independence - The values of the dependent variable must be independent of one another.
In other words, there should be no correlation between consecutive error terms, and each of the errors should have an average of 0.
Zero Conditional Mean - The X variable is exogenous, meaning it is uncorrelated with the error term.
This is known as the assumption of exogeneity. When this assumption is met, we say that the error has a zero mean or is orthogonal to X, which implies that E(u|X)=0.
The BLUE stands for Best Linear Unbiased Estimators. When OLS meets these five assumptions, the least-squares estimators of α and β are the BLUE.
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Which of the following is an example of a quantitative variable?
a. number of car in college per day
b. Gender of students
c. college major
d. type of favored food
The correct answer we get from the Quantitative variable. i.e. "a. number of car in college per day".
The correct answer is "a. number of car in college per day".
Quantitative variable:
The quantitative variable is used to indicate a variable that takes on numerical values. It is a type of statistical variable that is numeric and uses an objective measure to determine its value.
The variable measures the count, amount, or size of something.
To find out which of the given options is an example of a quantitative variable, we need to understand what it is.
So, the given options are:
a. number of cars in college per day. (Quantitative variable)
b. Gender of students. (Non-quantitative variable)
c. College major. (Non-quantitative variable)
d. Type of favored food. (Non-quantitative variable)
Thus, option a. number of cars in college per day is an example of a quantitative variable.
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add me on imessage 3377074503
if i was you i wouldnt do that that could be inaporpiet
I have 10 shirts, 9 pants, 4 shoes, and 1 hats. How many unique outfits do I have?
The integral f 5/1-4a²da is to be evaluated directly and using a series approximation. a) Evaluate the integral exactly, using a substitution in the form ax = sin and the identity cos²x = (1 + cos2x). Enter the value of the integral: 1.96 b) Find the Maclaurin Series expansion of the integrand as far as terms in 2. Give the coefficient of in your expansion: -10.0 Submit part c) Integrate the terms of your expansion and evaluate to get an approximate value for the integral. Enter the value of the integral: 4 d) Give the percentage error in your approximation, i.e. calculate 100x (approx answer - exact answer)/(exact answer). % Enter the percentage error
The value of the integral using substitution is 1.96 and using approximation it can be truncated as it is an infinite series.
We need to evaluate the integral f(5/1-4a²) da directly and using a series approximation.
Substitution Method:
Consider ax = sin, then a = sin(x)/x.
On differentiating,
da/dx = cos(x)/x - sin(x)/x²; da/dx = 1/x cos²x.
We know that
1-4a² = 1-4(sin²x)/(x²) = cos(2x)/x²
So the given integral f(5/1-4a²) da can be written as
f (5/x² cos²x cos(2x)) dx = 5 f (cos(2x)/x²) dx
Integrating w.r.t x,
5 f (cos(2x)/x²) dx = (5/2) f (cos(2x)/x²) + C
Using the given values, we can find the value of integral which is 1.96. Thus, the answer is 1.96.
Approximation Method:
As we need to use a series approximation, we can use Taylor series for approximating the integral, which is:
f(x) = f(a) + (x - a) f'(a) + (x - a)² f''(a) / 2 + ... ∞
We know that the function f(5/1-4a²) can be written as
f(x) = 5 / (1 - 4x²), where a = 0.
Substituting the value of f(x) and f'(x) in the Taylor series formula, we get:
f(x) = f(0) + xf'(0) + x²f''(0) / 2! + x³f'''(0) / 3! + ...
The first derivative is given by:
f'(x) = -40x / (1 - 4x²)²
Substituting the value of f(0) = 5 and f'(0) = 0 in the Taylor series formula, we get:
f(x) = 5 - 40x² + 320x^4 - ...
This is an infinite series. Hence, we can truncate it after a few terms to get an approximate value of the integral. Thus, we can find an approximation for the given integral.
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Obtain the Inverse Laplace Transforms of: i. F(S) = (s+1)(s+5)
The inverse Laplace transform of F(S) = (s + 1)(s + 5) is δ''(t) + δ'(t) + 5δ(t), where δ(t) represents the Dirac delta function.
To obtain the inverse Laplace transform of F(S) = (s + 1)(s + 5), we can use the linearity property and the inverse Laplace transform table.
The table states that the inverse Laplace transform of s + a is equal to the Dirac delta function δ(t - a), and the inverse Laplace transform of a constant times F(S) is equal to the constant times the inverse Laplace transform of F(S).
Applying these properties, we can write:
F(S) = (s + 1)(s + 5)
= s^2 + 6s + 5
Using the linearity property, we can split this expression into three terms:
F(S) = s^2 + 6s + 5 = s^2 + s + 5s + 5
Taking the inverse Laplace transform of each term separately, we have:
L^(-1){s^2} + L^(-1){s} + 5L^(-1){s} + 5L^(-1){1}
Referring to the inverse Laplace transform table, we find that:
L^(-1){s^2} = δ''(t)
L^(-1){s} = δ'(t)
L^(-1){1} = δ(t)
Therefore, the inverse Laplace transform of F(S) = (s + 1)(s + 5) is:
L^(-1){F(S)} = δ''(t) + δ'(t) + 5δ(t)
In summary, the inverse Laplace transform of F(S) = (s + 1)(s + 5) is given by δ''(t) + δ'(t) + 5δ(t).
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I really need this answer right away. Im taking an exam.
How long has the polygraph machine been in use?
A.Over 2000 years
B. Over 1000 years
C.Over 200 years
D. Over 100 years
Answer: 100 years
Step-by-step explanation:
Find the sum of all angles of the quadrilateral if sides are in ratio 2:3:6:9
2 =
3 =
6 : 9 =
Step-by-step explanation:
Here are some solutions! Hope this helps.
-Lei
Can someone help me I really don’t understand this..
Answer:
this would be the answer 4.66666667 but when rounded it would be 5 or 4.7
Step-by-step explanation:
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During the summer Tyler delivers packages a Miranda Cares for animal on a farm to show there earnings Tyler makes a graph and Miranda makes a table, which statement is true?
Answer:
divide the earnings number by the time, and then subtract the quotient by the slope of tylers graph.
You would like to lease a car worth $36,601 for a three-year period. The leasing company told you that after three years, the car would have a residual value of $15,818. How much of the car’s original value will you have to pay off during your three-year lease?
Answer:
$20,783
Step-by-step explanation:
The original value of the car is given as $36,601.
Now, the residual value of the car after a 3 - year lease is given as $15,818.
Residual value is the amount that the person leasing this car can purchase it after the 3 years lease is expired.
Formula for his residual value is usually;
Residual value = original value - depreciation value
Where depreciation value is the amount of the cars original value the Person getting the vehicle on lease for 3 years would pay to cover that period.
Thus;
Depreciation value = 36601 - 15818 = $20783
The following represents a sequence with an exponential pattern. FInd the missing numbers of the sequence.
20, __, __, __, 12, 500
The complete sequence is:
20, 0.6, 41.6667, 69.4444, 12, 500.
To find the missing numbers in the given sequence, let's analyze the pattern and fill in the gaps.
Looking at the sequence, we can observe that there is an exponential pattern occurring.
Let's break it down step by step:
Starting with the first number, 20, we can see that each subsequent number is decreasing or increasing exponentially.
If we divide the second number by the first number, we get 12/20 = 0.6. Similarly, dividing the third number by the second number gives us 500/12 = 41.6667.
We can see that there is a decreasing exponential pattern from the first to the third number.
Let's continue this pattern:
If we divide the fourth number by the third number, we get an approximate value of 41.6667/0.6 = 69.4444.
So, we can deduce that the missing numbers in the sequence are approximately 0.6, 41.6667, and 69.4444.
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what is the value of y
Answer:
y= -6
Step-by-step explanation:
-2y - 5y= -7y
-7y=42
42/-7=
y= -6
Factor the difference of perfect squares. 4x^2 - 1
Answer:
(2x+1)(2x-1)
Step-by-step explanation:
4x^2-1
(2x)^2-1
(2x)^2-1^2
(2x+1)(2x-1)
For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. A random sample of 5724 physicians in Colorado showed that 3305 provided at least some charity care (i.e., treated poor people at no cost). in USE SALT (a) Let p represent the proportion of all Colorado physicians who provide some charity care. Find a point estimate for p. (Round your answer to four decimal places.) three decimal places.) (b) Find a 99% confidence interval for p. (Round your answers lower limit upper limit Give a brief explanation of the meaning of your answer in the context of this problem. We are 1% confident that the true proportion of Colorado physicians providing at least some charity care falls above this interval. We are 99% confident that the true proportion of Colorado physicians providing at least some charity care falls outside this interval. We are 99% confident that the true proportion of Colorado physicians providing at least some charity care falls within this interval. We are 1% confident that the true proportion of Colorado physicians providing at least some charity care falls within this interval. (c) Is the normal approximation to the binomial justified in this problem? Explain. No; np > 5 and ng < 5. Yes; np < 5 and ng < 5. Yes; np > 5 and ng > 5. No; np < 5 and ng > 5.
(a) The point estimate for p is 0.5763.
(b) The 99% confidence interval for p is (0.5645, 0.5882).
(c) The normal approximation to the binomial can't be used in this problem.
(a) Let p represent the proportion of all Colorado physicians who provide some charity care.
Find a point estimate for p.
From the given data, the proportion of Colorado physicians who provide charity care (p) is:
p = 3305 / 5724= 0.5763
The point estimate for p is 0.5763 (rounded to four decimal places).
(b) Find a 99% confidence interval for p.
A 99% confidence interval for p is given by:
p ± z * sqrt[p(1-p)/n], where n = 5724, z = 2.576 (for a 99% confidence level)
Lower limit = 0.5645 Upper limit = 0.5882
The 99% confidence interval for p is (0.5645, 0.5882).
We are 99% confident that the true proportion of Colorado physicians providing at least some charity care falls within this interval.
(c) No; np > 5 and ng < 5.
In general, the normal approximation to the binomial is valid when np and n(1-p) are both at least 5.
Here, n = 5724 and p = 0.5763.
So, np = 5724 × 0.5763 = 3303.9620
≈ 3304n(1-p)
= 5724 × (1 - 0.5763)
= 2419.0380
≈ 2419
Neither np nor n(1-p) is less than 5.
So, the normal approximation to the binomial can't be used in this problem.
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Find the area of the polygon.
Answer:
120
Step-by-step explanation:
The polygon shown is a parallelogram with a base length of 15 and a height of 8
The area of a parallelogram can easily be calculated by multiplying the height by the base length
So A = 15 * 8
15 * 8 = 120
Hence, the area of the polygon is 120 square units
Will give brainless whatever that means. Due today. The radius of a basketball is 10 inches, and the radius of a soccer ball is 9 inches. How much greater is the volume of the basketball than the soccer ball? Round to the nearest tenth
9514 1404 393
Answer:
1135.2 in³
Step-by-step explanation:
The volume of a sphere is given by the formula ...
V = (4/3)πr^3
The difference in volume will be ...
∆V = Vb -Vs = (4/3)π(10^3 -9^3) = (4/3)π(1000 -729) = 4/3π·271 . . . cubic in
∆V ≈ 1135.2 . . . cubic inches
The basketball has a volume that is 1135.2 cubic inches greater.
_____
If you use 3.14 for π, your answer is 1134.6 in^3.
Pls help will mark brainliest
Answer:
9) (68 + 160)/2 = 114°
10) 80/2 = 40°
11) 105 x 2 = 210°
12) 80 x 2 = 160°
13) (360 - 180 - 108)/2 = 36°
14) (360 - 69 - 95)/2 = 98°
Help me with this please
Answer:
21 degrees.
Step-by-step explanation:
Answer:
21
Step-by-step explanation:
as BC ║DE ⇒ ∠CBA=∠BAD
⇒ ∠x=180-∠BAD -∠CAB
⇒ ∠x=180-105-54=21
Is it possible to draw a quadrilateral that is not a rectangle, but has at least one right angle.Explain
Answer:
Yes
Step-by-step explanation:
A trapezoid with two right angles is a quadrilateral with two right angles.
You deposited $700 in a savings account 4 years ago. The simple interest rate you earn is 4%. What is the amount of interest that you earned in those 4 years?
Answer: $112
Step-by-step explanation:
700x0.04=28x4=112
Falls City has a current population of 3,300 and is increasing 4.1% every year. How large will the population be in 8 years? Round your answer to the nearest whole number.