there may be several different min-cut sets in a graph. using the analysis of the randomized min-cut algorithm, argue that there can be at most n(n − 1)/2 distinct min-cut sets.
The randomized min-cut algorithm, such as the Karger's algorithm, is an iterative algorithm that repeatedly contracts edges in a graph until only two nodes (or a small number of nodes) remain. At that point, the remaining edges represent a cut in the graph.
In each iteration of the algorithm, an edge is chosen uniformly at random to be contracted. This contraction merges the two nodes connected by the chosen edge into a single super-node. The process continues until only two nodes remain, representing the cut in the graph.
To analyze the algorithm, let's consider a graph with n vertices. At each iteration, the number of vertices decreases by one since two vertices are merged into one. Therefore, after k iterations, there are n - k vertices remaining in the graph.
Now, let's consider the number of distinct cuts that can be formed by the remaining vertices. For n vertices, the total number of possible cuts is [tex]2^(n-1)[/tex]since each vertex can be on one side of the cut or the other. However, some of these cuts may be identical because the order in which the vertices are contracted can change the representation of the cut.
To see why, suppose we have a set of vertices A and a set of vertices B. The order in which the vertices are contracted can result in different representations of the cut. For example, if we contract vertex A before vertex B, the cut might be represented as (A, B). However, if we contract vertex B before vertex A, the cut might be represented as (B, A). Both cuts are essentially the same, but the order of the vertices determines the representation.
Since there are (n-1) edges that need to be contracted to reach the final cut of two vertices, there are (n-1)! possible orders in which the vertices can be contracted. However, each order produces the same cut, so we need to divide by (n-1)! to account for the different representations.
Therefore, the number of distinct cuts that can be formed by the remaining vertices is [tex]2^(n-1)[/tex]/ (n-1)!. Simplifying this expression, we get:
[tex]2^(n-1) / (n-1)! = n(n-1)(n-2)...(2)(1) / (n-1)(n-2)...(2)(1) = n[/tex]
So, there can be at most n distinct min-cut sets in the graph.
In summary, using the analysis of the randomized min-cut algorithm, we can argue that there can be at most n(n - 1)/2 distinct min-cut sets.
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Prove that if A is a proper nonempty subset of a connected space X, then Bd(A) +0.
If A is proper "nonempty-subset" of "connected-space" X, then boundary of A, is nonempty because every point in A is either interior or exterior point.
In order to prove that if A is proper "nonempty-subset" of "connected-space" X, then boundary of A, which is denoted Bd(A), is nonempty, we proof this by contradiction.
We assume that A is proper "nonempty-subset" of "connected-space" X, and suppose, that Bd(A) is empty,
Since Bd(A) is set of all "boundary-points" of A, the assumption that Bd(A) is empty implies that there are no "boundary-points" in A,
If there are no "boundary-points" in A, it means that "every-point" in A is either an "interior" or "exterior-point" of A,
Consider the sets U = A ∪ X' and V = X\A, where X' represents the set of exterior points of A. Both U and V are open sets since A is a proper nonempty subset of X.
U and V are disjoint sets that cover X, i.e., X = U ∪ V,
Since X is a connected space, the only way for X to be written as a union of two nonempty disjoint open sets is if one of them is empty. Both U and V are nonempty since A is proper and nonempty.
So, the assumption that Bd(A) is empty leads to a contradiction with the connectedness of X.
Thus, Bd(A) must be nonempty when A is a proper nonempty subset of a connected space X.
By contradiction, we have shown that if A is a proper nonempty subset of a connected space X, then the boundary of A, Bd(A), is nonempty.
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The given question is incomplete, the complete question is
Prove that if A is a proper nonempty subset of a connected space X, then Bd(A) ≠Φ.
HELP ME UNSCRAMBLE THESE WORDS PLEASE!!! IGNORE THE SUBJECT I JUST DIDN'T KNOW WHAT TO PICK!! THANK YOU LOVELIES!!!
1) Play
2)Dribble
3)Paint
4)Center
5)Assists
6)Perimeter
7)Benchwarmer
8)Aboard
9)Technical
10)Rebound
11)Penalty
12)screen
13)Overtime
14)guard
15)? Host
16)Quarter
17) Bounce
18)Swish
19)Halftime
20)Backcourt
A function whose graph goes down (falls) as it is followed from left to right is said to be a ____ function.
first two letters de
Bill has 29 more apps on his phone than Sherri, and they have a total of 99 apps. How many apps does each person have?
Sherri has 35 apps on her phone and Bill has 64 apps on his phone.
Let's represent Sherri’s apps with x.
Then, the number of Bill's apps will be x+29 (since he has 29 more apps than Sherri).
Their total number of apps is 99.Thus, the mathematical equation is:x + (x+29) = 99
Simplifying this equation gives:2x + 29 = 99
Subtracting 29 from both sides of the equation gives:2x = 70
Dividing both sides by 2 gives:x = 35
This means Sherri has 35 apps on her phone.
Substituting that into x+29 gives:35+29 = 64
Therefore, Bill has 64 apps on his phone.
Hence, each person has Sherri has 35 apps on her phone and Bill has 64 apps on his phone. The total number of apps between the two of them is 99 apps.
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(Past Due) Need Help
I would say its the second one.
the student shouldve distributed the 2^3x+3 into 2^3x+9
What is another way to write -8+5?
Answer:
5-8
Step-by-step explanation:
you just had to move the 8 over
Let L = {w ∈ {a, b}^∗| w has twice as many a′s as b′s}. Draw the state diagram of a P DA that accepts language L. Your P DA should not be overly complicated.
The transitions are labeled with the input symbol, the symbol to be pushed onto the stack (ε indicates no symbol is pushed), and the symbol to be popped from the stack (ε indicates no symbol is popped).This PDA accepts strings in L where the number of 'a's is twice the number of 'b's.
To draw the state diagram of a PDA that accepts the language L = {w ∈ {a, b}^∗ | w has twice as many a's as b's}, we can design a simple PDA with two states.
State 1: Initial state
Transition: (a, ε, a) -> State 1
Transition: (b, a, ε) -> State 2
Transition: (ε, ε, ε) -> Accepting state
State 2: Secondary state
Transition: (b, a, ε) -> State 2
Transition: (ε, ε, ε) -> Accepting state
Accepting state: Final state to indicate that the input string is accepted.
Here is the state diagram representation of the PDA:
Note: Find the attached image for the state diagram representation of the PDA.
In this PDA, State 1 is the initial state, and State 2 is the secondary state. The transitions are labeled with the input symbol, the symbol to be pushed onto the stack (ε indicates no symbol is pushed), and the symbol to be popped from the stack (ε indicates no symbol is popped).
The PDA works as follows:
In State 1, for each 'a' encountered, no symbol is pushed onto the stack, and the PDA remains in State 1.In State 1, for each 'b' encountered, 'a' is pushed onto the stack, and the PDA transitions to State 2.In State 2, for each 'b' encountered, 'a' is popped from the stack, and the PDA remains in State 2.If the input string is consumed and the PDA is in State 1 or State 2, it transitions to the accepting state.This PDA accepts strings in L where the number of 'a's is twice the number of 'b's.
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What solid figure is shown below?
Answer:
rectangular prisim
Step-by-step explanation:
Answer:
rectangular prism
Step-by-step explanation:
rectangle shaped box
how do I solve this??
Answer:
i guess you multiply 3and 30good lu8ck
Step-by-step explanation:
Answer:
3x+30=90 complementary
3x=90-30
x=60/3=20
x=20° is your answer
Ben uses 3/12 pound of strawberries and 2/12 pound of blueberries to make jam. How many pounds of berries does Ben use to make jam?
Answer:
5/12
Step-by-step explanation:
add 3/12 and 2/12 to get 5/12 (decimal form is .416)
Answer:5/12 pounds
Step-by-step explanation: You take the 3 and add it with the 2 and you get 5/12
[tex]\frac{3}{12}[/tex] + [tex]\frac{3}{12}[/tex] = [tex]\frac{5}{12}[/tex]
A window in the shape of a semi circle has a radius of 40 cm. The window is shown below. Find the area of the window.
Answer:
[tex]A=2513.27\ cm^2[/tex]
Step-by-step explanation:
The radius of semicircle window, r = 40 cm
The area of semicircle is given by :
[tex]A=\dfrac{\pi r^2}{2}[/tex]
Substitute all the values in the above formula.
[tex]A=\dfrac{\pi \times 40^2}{2}\\\\A=2513.27\ cm^2[/tex]
So, the area of the window is equal to [tex]2513.27\ cm^2[/tex].
The following table shows the number of miles (d) a car travels in t hours while driving at a constant speed of 55 miles per hour.
t 1 2 3 4 5
d 55 110 165 ? 275
How many miles will the car travel in 4 hours?
Answer: 220 miles
Step-by-step explanation:
Car goes 55 miles in 1 hour.
Constant speed.
Formula is 55t = d.
55 x 4 = 220 miles
Answer:
the guy above is right
Step-by-step explanation:
PLSSS HELP IMMEDIATELY!!! ILL GIVE BRAINIEST!!! (if u provide a link, i’m not giving u brainiest!)
Answer: B. (2,1)
Step-by-step explanation:
its not c because if you look at R you see that it is going 4,3 not 3,4 because when you doing ratios on a number line it goes x-axis then y-axis next not y-axis then the x-axis.
It's not D because if you look at Q you see that it is going 3,5 not 5,3 because when you doing ratios on a number line it goes x-axis then y-axis next not y-axis then x-axis.
That's the same thing for A so your answer is b.
Answer:
B.......................
How to do this with steps please help !
Answer:
126+111+43+8=288
360-288=72
72÷12=6
x=6
Answer:
x=6
Step-by-step explanation:
Step by step explaination attached below.
Hey I'm Chloe Can you Help Me I will give Brainlest, Thank you :)
Pythagoras lived over 2500 years ago. What is his theorem and why do we still use it today?
Answer:
The Pythagorean Theorem is helpful for two-dimensional navigation. You can use it along with two lengths to calculate the shortest path. The lengths north and west will be the triangle's two wings, and the diagonal will be the shortest line separating them. The same principles can be used for air navigation. He is best known in the modern day for the Pythagorean Theorem, a mathematical formula which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.
- Hope this helps! :)
PLEASE HELP ITS DUE TODAY!!
For problems 8-11, write Yes or No whether each figure is a polygon. (1 point each)
Answer:
yes,yes,yes,no :) hope it helps
Step-by-step explanation:
Answer:
1. Yes
2. No
Definition of a Polygon: a plane figure with at least three straight sides and angles, and typically five or more.
Last year, Jackson bought a brand new car for $47,500. If the car depreciates in value by 20% each year, what will the car be worth when it is 8 years old?
Answer:
exponential growth or exponential decay , and what ... The value of a car purchased for $20,000 decreases ... population, P, increases by 20%each year,
what is the distance between (3,-5), (-3,0)
Answer: 11 units
Step-by-step explanation:
-3 and 3 are 6 units apart, so that would be 6 units, and when you add the -5 into the problem, the 6 units become 11 units. Hope this helped :)
Answer:
Exact Form:
√ 61
Decimal Form:
7.81024967 …
The pathway of a frog jumping onto a lily pad can be represented by the equation h= -0.5t^2 +3t+2
Answer:
h = 6.5 feet
Step-by-step explanation:
The height of the frog as a function of time is given by :
[tex]h= -0.5t^2 +3t+2[/tex] .....(1)
We need to find the maximum height reached by the frog. We can find it as follows :
Put [tex]\dfrac{dh}{dt}=0[/tex]
So,
[tex]\dfrac{d}{dt}(-0.5t^2 +3t+2)=0\\\\-t+3=0\\\\t=3[/tex]
Put t = 3 in equation (1).
[tex]h= -0.5(3)^2 +3(3)+2\\\\h=6.5\ feet[/tex]
So, the maximum height is 6.5 feet.
PLEASE HELP!! DON'T JUST TAKE POINTS :(
The wingspan of a hawk is the distance from the end of one spread-out wing to the end of the other spread-out wing. A scientist measured the wingspans of a random sample of hawks at a national park. Based on the median wingspan of the sample, the scientist estimates that the median wingspan of all hawks in the national park is 40 inches. Which graph most likely represents the data from the scientist's sample?
Answer:
I think the most upright answer would be D
Step-by-step explanation:
You are conducting research using rhesus macaques (monkeys). For ethical reasons as well as limited resources, you decide to use only 9 animals, 3 animals in each of 3 treatment groups. To increase the power of your statistical testing, you take 10 samples from each animal. What statistical test might you use in this instance?
Your graduate student objects, saying that taking multiple samples from each animal cannot increase statistical power. How should you respond?
The response to your graduate student's objection of taking multiple samples from each animal cannot increase statistical power is:
You should respond by saying that taking multiple samples from each animal is a well-known strategy for increasing statistical power.
In addition, increasing the number of observations per group improves the statistical test's accuracy and reliability.
The statistical test that one would use in this instance is the One-way ANOVA, or one-factor ANOVA.
A statistical test that can be used in this instance is the One-way ANOVA, or one-factor ANOVA.
This statistical test is a method used to determine if the average or mean of a numerical variable varies significantly between two or more groups of interest.
For this statistical test, it is best to use multiple samples to increase statistical power.
In addition, ANOVA is used to compare three or more sets of data for statistical significance by testing for variances.
ANOVA's null hypothesis is that all populations are equal, while the alternative hypothesis is that at least one population is different from the others.
The response to your graduate student's objection of taking multiple samples from each animal cannot increase statistical power is:
You should respond by saying that taking multiple samples from each animal is a well-known strategy for increasing statistical power.
In addition, increasing the number of observations per group improves the statistical test's accuracy and reliability.
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Question 8 of 10 A differential equation is: A. any equation involving a differentiable function. B. any equation involving an integral function. C. any equation involving a derivative. D. any equation involving two or more derivatives. E. any equation involving a derivative where the antiderivative is known.
A differential equation is an equation involving a differentiable function, which is a critical tool in modeling physical phenomena like population growth, radioactive decay, and fluid flow.
A differential equation is an equation that involves a differentiable function. It is an equation in which the variables' derivatives appear. Differential equations are used to model physical phenomena like population growth, radioactive decay, and fluid flow. The order of a differential equation is the highest order of the derivative of the function. A first-order differential equation has the highest order of 1, and a second-order differential equation has the highest order of 2.A differential equation can be classified into three types: Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs), and Differential Algebraic Equations (DAEs). Ordinary differential equations have a single independent variable and one or more dependent variables that depend on it. Partial differential equations have more than one independent variable and multiple dependent variables that depend on each other. Differential algebraic equations have both derivatives and algebraic equations in them.A differential equation is essential in physics, engineering, and mathematics. It is used to model many natural phenomena and helps in predicting the future. Most differential equations can not be solved analytically, so numerical methods are used to find approximate solutions. In conclusion, A differential equation is an equation involving a differentiable function, which is a critical tool in modeling physical phenomena like population growth, radioactive decay, and fluid flow.
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Please help!!!I’ll mark you as brainliest!!!!
Round 0.206896552 to the nearest tenth as a percentage
Answer:
20%
Step-by-step explanation:
Answer:
20%
Step-by-step explanation:
0.206896552 rounded to the nearest tenth is 0.2 (The 0 in the hundredths place rounds down)
To find 0.2 as a percentage simply multiply it by 100
0.2*100=20%
To buy tickets online for a circus, there is a one-time processing fee of $5 and each
ticket costs $65. If Winston is buying tickets for himself and up to 3 of his friends, then
which statement below best represents this situation?
A. The domain is {70, 135, 200, 265).
B. The range is the total price of the tickets purchased and includes all whole numbers
from 1 to 4.
C. The range is {65, 130, 195).
D. The range is the total price of the tickets purchased and spans from 70 to 265.
I think the answer would be B. The range is then total price of the tickets purchased and includes all whole numbers from 1 to 4.
2. Find the inverse Laplace transform. ( 3pts each) 2 1 4 a. F(s) b. F(s) - 52 -28-3 S S
Inverse Laplace transform of f(s) = s³ / (s² + 6s + 13) is
[tex]f(t) = [(-3 + 2i)^{13} / (2i)] e^{(-3 + 2i)t} + [(-3 - 2i)^{13} / (-2i)] e^{(-3 - 2i)t[/tex]
The inverse Laplace transform of f(s) = s¹³ / (s² + 6s + 13) needs to be found.
To find the inverse Laplace transform, we first need to factor the denominator of f(s) using the quadratic formula:
s² + 6s + 13 = 0
s = [-6 ± √(6² - 4(1)(13))] / 2(1)
s = -3 ± 2i
Now we can rewrite f(s) as:
f(s) = s¹³ / [(s + 3 - 2i)(s + 3 + 2i)]
Using partial fraction decomposition, we can write:
f(s) = A / (s + 3 - 2i) + B / (s + 3 + 2i)
where A and B are constants to be determined. Multiplying both sides by the denominator, we get:
s¹³ = A(s + 3 + 2i) + B(s + 3 - 2i)
Substituting s = -3 + 2i, we get:
(-3 + 2i)¹³ = A(2i)
Solving for A, we get:
A = (-3 + 2i)¹³ / (2i)
Similarly, substituting s = -3 - 2i, we can solve for B:
B = (-3 - 2i)¹³ / (-2i)
Now we can write f(s) as:
f(s) = [(-3 + 2i)¹³ / (2i)] / (s + 3 - 2i) + [(-3 - 2i)¹³ / (-2i)] / (s + 3 + 2i)
Taking the inverse Laplace transform of each term separately using the table of Laplace transforms, we get the final answer:
[tex]f(t) = [(-3 + 2i)^{13} / (2i)] e^{(-3 + 2i)t} + [(-3 - 2i)^{13} / (-2i)] e^{(-3 - 2i)t[/tex]
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The given question is incomplete, the complete question is below
Find the inverse Laplace transform. f(s) = s¹³ / (s² + 6s + 13)
help me with this plesse
Choose the equation that best describes the situation below.
The Houston Rockets scored 66 points in the second half. There are 24 minutes in a half. What was their average number of points per
minute?
p=points per minute
That would just be
66/24, as there is 24 minutes in a half
66/24=2.75 points
p = 2.75 points per min
Please answer correctly! I will Mark you Brainliest!
Answer:
I think the volume of the figure is 60 but I'm not 100 % sure
Step-by-step explanation:
I belive the formula for this figure was (a+b)xh/(2)
So 40x 3 = 120
120/2 = 60
An object located at the point (4, 55) on a distance time graph is later located at the point (9, 45). If distance is in metres and time is in seconds, the average speed is: a) -2 m/s b) 5 m/s c) -5 m/s d) 2 m/s
An object located at the point (4, 55) on a distance time graph is later located at the point (9, 45). If distance is in metres and time is in seconds, the average speed is -2 m/s.
To calculate the average speed of an object, we need to find the total distance traveled divided by the total time taken.
Given that the object is located at the point (4, 55) initially and later located at the point (9, 45), we can determine the total distance traveled and the total time taken.
Total distance traveled = Difference in distance = 45 - 55 = -10 meters (negative because the object moved from a higher distance to a lower distance)
Total time taken = Difference in time = 9 - 4 = 5 seconds
Average speed = Total distance traveled / Total time taken = -10 meters / 5 seconds = -2 meters/second
Therefore, the average speed of the object is -2 m/s.
Option (a) -2 m/s is the correct answer.
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