A tree is called central if its center is K₁ and bicentral if its center is K₂. Show that every tree is either central or bicentral without using the theorem that the center of every connected graph G lies in a single block.

Recall that a set B is dense in R if an element of B can be found between any two real numbers a < b. Take p∈Z and q∈N in every case. It is given that the set of all rational numbers p/q with 10|p| ≥ q is not dense in R. Explain, using plain words (without a rigorous proof), why this is. That is, present a general argument in plain words. Does this set violate the Archimedean Property? If so, how? (PLEASE DON'T REPEAT THE ANSWERS TO THIS QUESTION ALREADY POSTED ON CHEGG)

On March 11, 2011, Japan suffered an earthquake and tsunami that caused a disastrous accident at the Fukushima nuclear power plant. Among many other results, amounts of iodine-131 that were 27 times the government limit were found in a sample of spinach 60 miles away.† Now, 27 times the government limit of iodine-131 is 54 thousand becquerels per kilogram.† The following table shows the amount I, in thousands of becquerels per kilogram, of iodine-131 that would remain after t days.

(a)

Show that the data are exponential. (In this part and the next, round to three decimal places.)

Because t increases by 1 each time, to show that the data are exponential, we must show that the successive ratios (rounded to three decimal places) are the same. Because all of the ratios are equal to  , the data are exponential.

(b)

Find an exponential model I that shows the amount of iodine-131 present after t days.

I(t) =

(c)

How long will it take for the amount of iodine-131 to fall to the government limit of 2 thousand becquerels per kilogram? Round your answer to the nearest whole day.

days

Verify the divergence theorem for the vector field F = 2xzi + yzj +z2k and V is the volume enclosed by the upper hemisphere x2 + y2 + z2 = a2, z ≥ 0

1. Solve for the optimal values of C1 and C2 in the following optimization problem: MaxC1,C2 C₁^1/2 + βC₂^1/2

s.t. C₁ + C₂ /1 + r = Y₁ + Y₂/1 + r

Hint: ∂C^1/2 /∂C = 1/2C^−1/2

When r goes up, how does C1 change? Does it increase or decrease?

Consider a region R bounded by the y-axis, the line segment y=8-x for x from 0 to 8, and part of the circle y=-sqrt(64-x^2) for x from 0 to 8. Find the centroid.

use well ordering principle to prove that sqrt(6) is not rational

A furniture company produces tables (X) and chairs (Y). Available resources have indicated that the production volume level should not exceed 1200 units per week and the demand for chairs is at most half of that for tables. Further, the production level of tables can exceed three times the production of chairs by at most 600 units. If the company makes a profit of OMR12 and OMR 16 per unit respectively on tables and chairs, how many of each should be produced weekly in order to maximize the profit. which of the following represents one of the constraints for the above business problem

Show that {xx^R | x,y ∈ {0,1}*} is a context-free language.

Note that x^R is the reversal of x.

Show all work.

Question is for Discrete Math Structures

Consider the nonhomogeneous equation y"-8y'+16y=e^4x cos x. Find a particular solution of the equation by the method undetermined coefficients.

Graph Theory

Prove:

If G is a graph for which deg(u)+deg(v) ≥n for each uv ∈EsubG, the G has a Hamiltonian cycle.

(with counter examples)

I'm stuck on this python exercise using plots

Consider the hyperbolic paraboloid function:

z=x^2−y^2

Create several plots of the hyperbolic paraboloid function. Use true aspect ratio and label all axes.

Any help would be appreciated!Matlab can be used as long as the code can be used in Python as well

Suppose the joint probability distribution of two binary random variables X and Y are given as follows.

X goes along side as 0 and 1, Y goes along top as 1 and 2

e) Find joint entropy H(X, Y ).

f) Suppose X and Y are independent. Show that H(X|Y ) = H(X).

g) Suppose X and Y are independent. Show that H(X, Y ) = H(X) + H(Y ).

h) Show that I(X; X) = H(X).

please help with all parts!

thank you!

Discret Math MATH/CSCI2112

(3) (a) Write the following premises and the conclusion as logical statements and prove the conclusion correct: (use the symbols in brackets).

If the flight is late, I will spend the night in Toronto. If I miss my flight I will spend the night in Winnipeg. Either my flight is not late, or I did not miss my flight from Winnipeg, but not both. Therefore either I will spend the night in Toronto, or I will spend the night in Winnipeg. (L, T, M, W)

(b) If the conclusion is changed to: "Either I will spend the night in Toronto or I will spend the night in Winnipeg, but not both" Is the conclusion still correct?

In this exercise we outline a proof of the following statement, which we will be taking for granted in our proof of the division theorem: If a, b ∈ Z with b > 0, the set

S = {a − bq : q ∈ Z and a − bq ≥ 0}

has a least element.

(a) Prove the claim in the case 0 ∈ S.

(b) Prove the claim in the case 0 ∈/ S and a > 0. (0 is not a member of S)

(c) Prove the claim in the case 0 ∈/ S and a ≤ 0. (0 is not a member of S)