# Tagged Questions

Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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### Simplify expression of conditional entropies to entropy rate

I have this expression $lim_{m \to \infty} \frac{1}{m}(\frac{m}{2}*H(X_{2i}) + \frac{1}{m}H(X_{2i+1}|X_{2i},X_{2i+2}))$ And I need to simplify this into something that resembles the entropy rate ...
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### Is $f$ $\mu/\nu$ integrable?

Let $\Omega=\Bbb R$ and the measures $\mu(A)=\lambda(A\cap\Bbb R_{<0})+\pi(A\cap\Bbb R_{<=0})$ and $\nu(A)=\lambda(A)+\pi(A\cap\Bbb Z_{<=0})$ where $\lambda$ is the Lebesgue-Borel measure and ...
1answer
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### Composite Relations - Give Examples of relations $R_1$ and $R_2$ such that $R_2 \circ R_1 = R_1 \circ R_2$ and $R_2 \circ R_1 \neq R_1 \circ R_2$

Let $S =[a,b,c]$. Give examples of a. relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 = R_1 \circ R_2$ b. relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 \neq R_1 \circ R_2$ My ...
1answer
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### Rudin Chapter 10 exercise 8 - Evaluating the integral

The question: Let $H$ be the parallelogram in $\mathbb{R}^2$ whose vertices are $(1,1), (3,2), (4,5), (2,4).$ Find the affine map $T$ which sense $(0,0)$ to $(1,1), (1,0)$ to $(3,2), (0,1)$ to ...
3answers
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### Exponential equation with one 'trouble' term

I have the following equation: $4^{2x} - 12 * 4^{x} + 32 = 0$ But this $12$ doesn't make me able to have all terms in the same base, how to proceed with it ?
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### Why is this not trivial? - $f(x)~|~g(x) \iff g(x) \in \langle f(x) \rangle$

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
1answer
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### Graceful Labeling for cycle

This result has been proved by Rosa, but I can't find a link to see his paper. I want to show that the graph $C_n$ is graceful if and only if $n=4k$ or $n=4k-1$ for some integer $k$. It's not hard to ...
1answer
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### Matrix rotation, projection, and reflection

What 3 by 3 matrices represent the transformations that a) project every vector onto the $x-y$ plane? b) reflect every vector through the $x-y$ plane? c) rotate the $x-y$ plane through 90 degrees, ...
3answers
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### Equality holds in triangle inequality iff both numbers are positive, both are negative or one is zero

How do we show that equality holds in the triangle inequality $|a+b|=|a|+|b|$ iff both numbers are positive, both are negative or one is zero? I already showed that equality holds when one of the ...
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### Recursive seqence of power set, starting with the empty set

Let $A_0$ be the empty set and $A_n := \mathcal{P}(A_{n-1})$ for $n \in \mathbb{N}$. I have to determine $A_n$ and $|A_n|$. Using the definition of the power set I get \begin{align} A_1 & = ...
2answers
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### First order differential equation unresolved over derivative

First order differential equation unresolved over derivative I really dont know what to do whit this equation, i have tried to integrate it, but i can do it. $$y´^3-yy´^2-x^2y´+x^2y=0$$
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### How to prove that in optimal strategie of tower of Hanoi none of the disks can't be placed on a disk of same parity

How to prove that in optimal strategie of tower of Hanoi none of the disks can't be placed on a disk of same parity if we have 3 rods. So for example disk 2 can't be placed on disk 4, or disk 1 can't ...
3answers
49 views

### Is this easy matrix diagonalizable or not?

Question Is $$\pmatrix{ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 }$$ diagonalizable if we allow only real numbers? if we ...
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135 views

### Maximizing the number of groups

The problem is as follows, There is a restaurant which has N number of chairs each chair has a unique number written on it so the array of chairs is like [1,2,....N-1,N] , there are G number of groups ...
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### Help figure examples of a flow on $\mathbb R^2$ where… [on hold]

Help figure examples where 1) $\omega (\overrightarrow x) = \emptyset$ for all $\overrightarrow x \in \mathbb R^2$ The closer thing I can think of is 0 vector...which does not work...please point in ...
1answer
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### counting the number of solutions

Let $p$ be a prime number. Count the number of solutions $A$ in $M_{n\times n}(\mathbb Z_p)$ to the equation $\det(A) = 0$. Attempt of solution: $\mathbb Z_p = \{0,1,2,...p-1\}$ I tried listing down ...
1answer
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### Addition of two cosine waves with different periods

I was just wondering if anyone knows how to add two different cosine equations together with different periods to form one equation. Is there a way to do this and get a real answer or is it just all ...
1answer
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### Conversion of bases with logarithms

The question says if $\log_6(2)$ is $a$ and $\log_5(3)$ is $b$, express $\log_5(2)$ in terms of $a$ and $b$. I have tried the change of base formula for $ab$ to no avail, can someone give me a hint ...
3answers
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### Proving or disproving uniform continuity of various real functions

Okay before anyone does anything, I don't want any of you guys to just write out proofs for all of these, that's asking a bit much :P Maybe just do one, and make it detailed because I really need ...
1answer
29 views

### Real integral using complex methods

Evaluate $\displaystyle\int_0^\infty \frac {x^\frac{1}{2}}{1+x^4}dx$ using complex methods. I'm totally locked up on this one and have thrown in the towel. My strategy was to integrate around a ...
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### Allocating observations to test whether two expected values are equal$\mu_{1}-\mu_{2}=\Delta\neq 0?$

Question: Suppose that you wish to test the hypothesis $H_{0}:\mu_{1}=\mu_{2}$ versus $H_{1}:\mu_{1}\neq \mu_{2}$,where both variances $\sigma^2_{1}$ and $\sigma^2_{2}$ are know ,A tatal of ...
1answer
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### Proving that a series converges to an irrational number.

I am taking honors Calculus II and have been doing reasonably well in the course until the current problem set which is due tomorrow. One exercise that is really giving me trouble is this: Prove ...
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### rolling fair 3-sided dice

Write down, with explanation, a polynomial which generates the number of ways to get each possible total from a single roll of a pair of standard fair three-sided dice.(Standard" means that the sides ...
0answers
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### Determining the joint probability distribution function of drawing three colored marbles at random.

The Question: Suppose an urn contains 10 marbles of which 7 are blue and 3 are red. Three marbles are to be selected from the urn one at a time, at random, without replacement. Let $X$ be the number ...
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### Proving a chain is aperiodic, and finding a stationary distribution.

We have an irreducible Markov chain with a not necessarily finite state space. It has a transition matrix $P$ such that $P^2=P$. Prove (1) the chain is aperiodic, and (2) prove $p_{ij}=p_{jj}$ ...
1answer
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### Rank, and existence of Matrices

Let $A$ be an $m\times n$ matrix. If the rank of $A$ is $m$, then prove there exists a matrix $B$, wich is $n \times m$, such that $AB=\text{I}_m$
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### Trig Identities: Finding the Pattern [on hold]

Can you find a polynomial in terms of $\cos (\theta)$ for $\cos (5\theta)$? $\cos (6\theta)$? Can you find a pattern so that $\cos (n\theta)$ could be written as a polynomial in cosine for any natural ...
1answer
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### Hypothesis testing question

The article “Statistical evidence of discrimination” (J. Ameri. Stat. Assoc., 1982, 773-83) discussed the court case of Swain v Alabama (1965), in which it was alleged that there was discrimination ...
1answer
56 views

### Question regarding $\epsilon-\delta-$proof

I want to prove the continuity of $f(x) = x^2$. Lets take $\epsilon > 0$ and $|x-x_0| <\delta$. I do: $$|f(x) - f(x_0) |= |x^2 - x_0^2| = |(x-x_0)(x+x_0)| < \delta |x-x_0|$$ Now the ...
1answer
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### A semi-generalization of the 2nd Borel-Cantelli lemma from Billingsley

I'm trying to prep for the PhD probability qual and I'm working through Billingsley. I've been struggling with question 4.11(b)/4.14(b) depending on your edition of the book: If for each k the series ...
2answers
263 views

### How would you prove Angles are Congruent/Equal?

I'm doing some Geometry work and I'm wondering how you would prove that two angles are congruent. I know that they are of the same angle size but I'm not sure why or how to prove it exactly.
1answer
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### Showing a substitution reduces a differential equation to a separable one

Suppose $M(x,y)dx + N(x,y)dy = 0$ is a homogeneous equation. Show that the substitution $x = vy$ reduces the equation to one with separable variable.