# All Questions

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### A poll carried a survey to examine the approval rate of a policy in a country. Which is most appropriate?

Suppose a polling company carried a sample survey to examine the approval rate of a newly implemented public policy in a country. 1,600 citizens aged 18 and above were randomly selected in this ...
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### how to integrate $\int \frac {x^4}{x^4 +5x^2 +4} dx$

Could I have some help for this question? $$\int \frac {x^4}{x^4 +5x^2 +4} \, dx$$ I've reduced the equation to... $$\int \frac {1}{1 + 5x^{-2} + 4x^{-4}} \, dx$$ But I'm stuck after this step. ...
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### Assume you toss a fair coin 25 times. How many completed runs do you expect to observe?

Assume you toss a fair coin 25 times with the outcome of each toss being independent of the outcomes of any other toss. How many completed runs do you expect to observe? By definition, completed ...
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### Bounding coefficients of a vector

I'm trying to prove the following lemma for a vector space: Suppose $X = span\{ e_1, ..., e_n \}$, with the $e_i$ linearly independent. (i.e. a basis, so you can uniquely write each $x\in X$ as a ...
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### Quick and Simple Real Analysis Bound

Given the inequalities: $$|f(x) - g(x)| < \epsilon\quad \forall \quad x \in [a,b]$$ and $$|g(x)| < M \quad \forall \quad x \in [a,b]$$ where $\epsilon > 0$ and $M > 0$. What is the ...
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### Breadth first search and bipartiteness

I was just wondering what the correlation is between a breadth-first search tree of a graph and that graph being bipartite?
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### Does the series $\sum_{n=1}^{\infty}\log n - (\log n)^{n/(n+1)}$ converge?

Does the series $$\sum_{n=1}^{\infty}\log n - (\log n)^{n/(n+1)}$$ converge?
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### Isomorphisms forced by series data of a group

I am interested in how far further than the Jordan-Holder theorem we can go. Say $1\triangleleft A\triangleleft B\triangleleft G$ and $1\triangleleft C\triangleleft D\triangleleft G$ are two ...
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### Remainder Theorem with polynomial as divisor

I'm aware that in remainder theorem you take the divisor and make it equal to zero. For the result of that just plug in x into whatever the polynomial dividend is. But its different when the divisor ...
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### Baby Rudin Chapter 3 Exercise 11(d)

I'm writing solutions to exercises in Baby Rudin I think might want to assign students, and I'm having particular difficulty with 11(d) in Chapter 3. Namely, for a sequence $(a_{n})_{n=1}^{\infty}$ of ...
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### Real Analysis 1 vs Real Analysis 2?

Note: I am not sure if this is the correct place to ask these types of questions, so please let me know if I should remove my question. I'm taking Real Analysis 1 this semester, and was thinking of ...
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### Proving $\lim\limits_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L$ implies $\lim\limits_{n\to \infty}\frac{a_n}{b_n}=L$

Suppose $(a_n)$ and $(b_n)$ are sequences where $b_n$ is increasing and approaching positive infinity. Assume that $\lim_{n\to \infty}$ $\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L$, where $L$ is a real number. ...
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### Which approach to follow: greedy, divide-n-conquer or dynamic programming?

Given any problem say we have to pick few objects out of N so that the total weight is below W considering all objects of SAME value, A variation for this problem can be to have values assigned to ...
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### Help with proof of chain rule

I'm following along on this proof of the chain rule. All is clear to me except the step where they say: Differentiablility implies continuity; therefore $\Delta_u \to 0$ as $\Delta_x \to 0$ in ...
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### Prove that recursive sequence is greater than 3/11

We just started learning sequences and my teacher gave us this problem that seems to be incredibly hard. I don't even know where to start. Question: The first term of the following sequence is $1$. ...
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### Poincare series

I understand the concepts used in then Poincare series, but I don't know how to compute the Poincare serie of an specific polinomyal, for example $x^2-a$, what must I do?
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C In parts 1-5 below, $G$ is a group and $H$ is a normal subgroup of $G$. Prove the following (Theorem 5 will play a crucial role) Theorem 5- Let G be a group and H be a subgroup of G. Then $(i)$ ...
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### Why the depth of this modules is 1?

I'm studying on these notes, my question in about a proof on page 63. Basically this is my question: Suppose $R$ local noetherian of positive depth, $M$ a module of finite gorenstein dimension ...
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### mathematical symbol for vector appending

Given a vector v=<1,2,3>, now I want to have a new vector v', which is the vector v, appends with a number 4, how should I represent v' mathematically? What I wish to have is something like ...
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### Probability of palindrome of length 11

What is the probability that a random bit string of length 11 is a palindrome? Anyone have an idea of how to do this? I'm not sure how to calculate the different cases.
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### What is the meaning of this operator?

The problem I'm solving involves a proof using the operator $ad A$, which is defined as follows: $ad$ $A$ $\cdot =[A,\cdot]$ What does this notation mean?
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### Prove whether or not the series converges? $\sum_{n=2}^{\infty} \frac{(-1)^n}{n^{1/n}}$

Any help is much appreciated on this!
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### 4 Points on Circumference of Circle and center

This is actually a computer science question in that I need to create a program that will determine the center of a circle given $4$ points on its circumference. Does anyone know the algorithm, ...
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### Positive integers less than $N$ not divisible by $4$ or $6$

How many positive integers less than $N$ are not divisible by $4$ or $6$ for some $N$?
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### A proof that the hexagonal lattice describes the optimal sphere packing in two dimensions

In an effort to understand sphere packing in 24 dimensions, I figured that I should start with 1 and 2 first. The proof of 1 dimension is obvious, since we can achieve 100% density, and you cannot do ...
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### A question on Intermediate Value Theorem

Suppose that $f$ is a continuous function and that $f(-1)=f(1)=0$. Show that there is $c\in(-1,1)$ such that $$f(c)=\frac{c}{1-c^2}$$ I am not sure if this is a new question as I set it this morning, ...
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### Question about definition of pullback as a smooth bundle map.

In Lee, there is an exercise involving the pullback that I can't understand. If $M,N$ are smooth manifolds and $F:M\to N$ is smooth, I am asked to show that the pullback $F^*$ is a smooth bundle map ...
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### Proving $\cos (z)$ is real for real values of $z$

Given the definition of cos function through the exponential function, how can we prove rigorously that for real values of $z$ $$\cos(z)=\operatorname{Re}(\exp(iz))$$?
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### “The Music of Pi” [closed]

For anyone interested, see the Oct/Nov MAA Monthly Supplement about The Music of Pi. I originally posted this along with a couple questions, but have since asked for the original question to be ...
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### Uniform convergence of difference quotients to the derivative

I remember being assigned the following homework problem a few years back. Let $f:[0,1] \to \mathbb{R}$ be continuously differentiable. Prove that, for every $\epsilon > 0$, there exists ...
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### what the likely size of the chance error if a simple random sample?

The National Assessment of Educational Progress periodically administers tests on different subjects to high school students. In 2000, the grade 12 students in the sample averaged 301 on the ...
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### Needing an example of one riemann integrable function

This is easy, but I couldn't find some example of a function that is not integrable but its Riemann improper integral exists and is finite
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### About the measurable subsets and the Lipschitz condition

I have, again, a doubt with the measurable subsets. If I have that $T\colon\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is Lipschitz, does $T$ send Lebesgue measurable sets in Lebesgue measurable sets. ...
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### About the properties of Lebesgue measurable subsets

This is a doubt about Lebesgue measurable subsets If I have two Lebesgue measurable subsets $E_1, E_2$ in $\mathbb{R}$, is the subset $E_1\times E_2$ Lebesgue measurable in $\mathbb{R}^2$?, If it ...
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### For any function f from a countable subset M of $\mathbb{R}$ there exists a sequence of continuous functions that converges pointwise to f on M.

Problem: Prove that for any function f from a countable subset M of $\mathbb{R}$ there exists a sequence of continuous functions {f_n} that converges pointwise to f on M. Context: This was put ...
This is a problem of Lebesgue measure and measure theory specifically. Suppose that $f:\mathbb{R}^2\longrightarrow [0,\infty)$ is measurable. $\Omega_1\subseteq \mathbb{R}^2$ is Lebesgue ...