Prove that if n is a perfect square then n = 4q or n = 4q...
Prove that if n is a perfect square then n = 4q or n = 4q + 1
for some q . Deduce that 1234567 is
not a perfect square. I don't understand why we get a=2q and a=4q+1
could we do the way like this n=a^2 , a=4q and a=4q+1 then
a^2=4(4)q^2 and 4(4q^2+2q)+1 as required is it ok? because the book
use a=2q and a=2q+1
Homework Answers
Consider a set of charges q, 2q, 3q, and 4q placed at the vertices of a...
Consider a set of charges q, 2q, 3q, and 4q placed at the
vertices of a square whose edge has a length a.
a. Calculate the total electric potential energy, U, that is
required to assemble this arrangement of charges compared with
having the charges infinitely far apart.
b. Now imagine that a charge Q is moved from infinitely far away
to the centre of this square of charges. Would it be possible to
assemble all five charges—Q and the...
Question 1) Ricky and Tom are doing math together. 1a) Rick thinks this is perfect square trinomial. Tom thinks this is...
Question 1) Ricky and Tom are doing math together. 1a) Rick thinks this is perfect square trinomial. Tom thinks this is a difference of two squares. Who is correct? [x2 + 8x + 16] · [x2 – 8x + 16] = (x2 – 16)2 1b) Rick thinks this is perfect square trinomial. Tom thinks this is a difference of two squares because the first term is negative. Who is correct? Why or why not? –9x2 – 30x – 25 1c)...
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that...
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that g(n) O(f(n)), or explain why one or the other is not true. (a) 2"+1 vs 2 (b) 22n vs 2" VS (c) 4" vs 22n (d) 2" vs 4" (e) loga n vs log, n - where a and b are constants greater than 1. Show that you understand why this restriction on a and b was given. f) log(0(1) n) vs log n....
9.14 Theorem. f the natural mumber N is a perfect square, then the Pell equation Ny 1 has no non-trivial integer soluti...
9.14 Theorem. f the natural mumber N is a perfect square, then the Pell equation Ny 1 has no non-trivial integer solutions. After all this talk about trivial solutions, let's at least confirm that in some cascs non-trivial solutions do cxist. 9.15 Exercise. Find, by trial and error at least two non-trivial solutions to each of the Pell equations x2-2y2 I and x-3y21 Rolstcred by the cxistence of solutions for N 2 and N 3, our focus from this point...
If we wanted to use the definition of isomorphism to prove that Z is not isomorphic to Q, we woul...
Abstract Algebra; Please write
nice and clear.
If we wanted to use the definition of isomorphism to prove that Z is not isomorphic to Q, we would have to show that there does not exist an isomorphism p : Z Q. In other words, we would have to show that every function that we could possibly define from Z to Qwould violate at least one of the conditions that define isomorphisms. To show this directly seems daunting, if not impossible....
I know that the sum of square of normal random variables follow a chi-square distribution. But...
I know that the sum of square of normal random variables follow a chi-square distribution. But when I learn how to do a goodness-fit test I don't know why the ratio of (O-E)^2/E follows a chi-square. I tried to square root of it first so that I might get something looks like a normal, but my new question arises : why (O-E)/sqrt(E) follows a normal-distribution? I know from sampling distribution that if the sample is from the same distribution as...
Problem 5 (Counting triominos) [20 marks] We saw in class that every 2 n × 2 n board, with one sq...
Problem 5 (Counting triominos) [20 marks] We saw in class that every 2 n × 2 n board, with one square removed, could be covered with triominos. Determine a formula counting the number of triominos covering such a truncated 2 n × 2 n board. Prove this formula by induction. I have seen solutions for this question posted but they seem to use iteration rather than induction and use notation that I don't understand.
I'm having trouble understanding the adiabatic expansion of a perfect gas. My book gives the following...
I'm having trouble understanding the adiabatic expansion of a
perfect gas. My book gives the following graph, but I don't
understand how the change in internal energy is 0 for process 1
(going from Ti,Vi to Ti,Vf). Youre expanding the gas-- would that
not require work, ie: loss of internal energy? It's not like heat
can be added to balance this out and keep the internal energy
constant, since this is adiabatic. Why does only the temperature
change (process 2)...
proof for distribution of (n-1)S^2/sigma^2 is the chi square distribution with n-1 degrees of freedom. I...
proof for distribution of (n-1)S^2/sigma^2 is the chi square
distribution with n-1 degrees of freedom.
I don't understand the expansion of the square, specifically how
certain terms disappeared and how a sqrt(n) appeared. Also towards
the end, why does V have a degree of freedom of 1? x A detailed
explanation of what happened from step 2 to step 3 would be very
helpful!
THEOREM B The distribution of (n − 1)S2/02 is the chi-square distribution with n – 1...
4. Here is a fact about permutations: (*) nPr= n!/(n-k)!, for all k =n. Let's prove...
4. Here is a fact about permutations: (*) nPr= n!/(n-k)!, for all k =n. Let's prove this via mathematical induction for the fixed case k-3. 2 of 3 (i) Write clearly the statement (**) we wish to prove. Be sure your statement includes the phrase "for all n" (ii) State explicitly the assumption in (*) we will thus automatically make about k-2 (ii) Now recall that to prove by induction means to show that IfmPm!/lm-k)! is true for all km...
Consider a set of charges q, 2q, 3q, and 4q placed at the
vertices of a square whose edge has a length a.
a. Calculate the total electric potential energy, U, that is
required to assemble this arrangement of charges compared with
having the charges infinitely far apart.
b. Now imagine that a charge Q is moved from infinitely far away
to the centre of this square of charges. Would it be possible to
assemble all five charges—Q and the...
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that g(n) O(f(n)), or explain why one or the other is not true. (a) 2"+1 vs 2 (b) 22n vs 2" VS (c) 4" vs 22n (d) 2" vs 4" (e) loga n vs log, n - where a and b are constants greater than 1. Show that you understand why this restriction on a and b was given. f) log(0(1) n) vs log n....
9.14 Theorem. f the natural mumber N is a perfect square, then the Pell equation Ny 1 has no non-trivial integer solutions. After all this talk about trivial solutions, let's at least confirm that in some cascs non-trivial solutions do cxist. 9.15 Exercise. Find, by trial and error at least two non-trivial solutions to each of the Pell equations x2-2y2 I and x-3y21 Rolstcred by the cxistence of solutions for N 2 and N 3, our focus from this point...
Abstract Algebra; Please write
nice and clear.
If we wanted to use the definition of isomorphism to prove that Z is not isomorphic to Q, we would have to show that there does not exist an isomorphism p : Z Q. In other words, we would have to show that every function that we could possibly define from Z to Qwould violate at least one of the conditions that define isomorphisms. To show this directly seems daunting, if not impossible....
I'm having trouble understanding the adiabatic expansion of a
perfect gas. My book gives the following graph, but I don't
understand how the change in internal energy is 0 for process 1
(going from Ti,Vi to Ti,Vf). Youre expanding the gas-- would that
not require work, ie: loss of internal energy? It's not like heat
can be added to balance this out and keep the internal energy
constant, since this is adiabatic. Why does only the temperature
change (process 2)...
proof for distribution of (n-1)S^2/sigma^2 is the chi square
distribution with n-1 degrees of freedom.
I don't understand the expansion of the square, specifically how
certain terms disappeared and how a sqrt(n) appeared. Also towards
the end, why does V have a degree of freedom of 1? x A detailed
explanation of what happened from step 2 to step 3 would be very
helpful!
THEOREM B The distribution of (n − 1)S2/02 is the chi-square distribution with n – 1...
4. Here is a fact about permutations: (*) nPr= n!/(n-k)!, for all k =n. Let's prove this via mathematical induction for the fixed case k-3. 2 of 3 (i) Write clearly the statement (**) we wish to prove. Be sure your statement includes the phrase "for all n" (ii) State explicitly the assumption in (*) we will thus automatically make about k-2 (ii) Now recall that to prove by induction means to show that IfmPm!/lm-k)! is true for all km...
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