0
votes
1answer
179 views

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 ...
1
vote
2answers
549 views

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. ...
2
votes
2answers
1k views

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 ...
3
votes
1answer
63 views

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 ...
3
votes
0answers
318 views

The number of injective functions

Find the number of injective functions (from finite set to finite set): $$ f:\{ 1,2,3,4,5,6,7\} \rightarrow \{ 1,2,3,4,5,6,7,8,9\} $$ with the following property: $$ f(i) \neq f(j)+1\ \text{ for ...
3
votes
1answer
120 views

Someone know a specific maximum principle…? [Solved]

I need a maximum principle in unbounded domains: if $u$ is a solution, bounded in $\Omega$, satisfying $$\Delta u+c(x)u=0, \ \ in \ \Omega,$$ $c\in L^\infty$, $$u\leq0 \ \ in \ \Omega$$ $$u(x_0)=0, \ ...
0
votes
1answer
33 views

Let $\Gamma$ be discrete in S. Then for any region $Ω$ in $S$, $Ω \cap \Gamma$ is discrete in Ω

I'm trying to understand this existence of triangulation's proof in this book. I have problems to understand the lemma 8.2.6: Let $\Gamma$ be discrete in S. Then for any region $Ω$ in $S$, $Ω \cap ...
2
votes
2answers
46 views

What's the solution to this system of equation?

$$ xy^3z^3 = yx^3z^3 = zx^3y^3 $$ Is there a way to solve this system? I think the answer is 1, but i can't verify my intuition.
4
votes
2answers
174 views

why the addition operation of a ring need to be commutative?

The definition of a ring requires the addition operation to be commutative. But why it has to be?
3
votes
3answers
262 views

Are trig identities commutative?

If I have $\cos(B)\cos(A)-\sin(A)\sin(B)$, can I write that as $\cos(A)\cos(B)-\sin(A)\sin(B)$? And then combine it as $\cos(A+B)$?
1
vote
3answers
195 views

Estimate sum of squares

If I know sum of some K numbers (each of them from 0 to 9). Can I find lower bound of sum of squares of the same elements? I don't know value of each element, I'm just aware of their sum. Thanks.
3
votes
1answer
96 views

Sanity check on spider web calculation

For fun I began reading an interesting online paper I found, Spiders for rank 2 Lie algebras, and on page $5$ we have the following calculation, akin to a tensor product expansion via bilinearity: ...
2
votes
1answer
1k views

Partial derivative of a summation.

I am trying to confirm a stated result on my lecture slide. Question: Given that $A:= \sum_i^n \frac{a_i}{(1+b)^{t_i}}$, where $a_i,b \in \mathbb{R}_+$ and $t_i \in \{t_1,...,t_n\}$ where $0 < ...
-1
votes
1answer
40 views

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 ...
0
votes
1answer
150 views

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?
1
vote
3answers
389 views

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?
1
vote
2answers
43 views

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 ...
2
votes
2answers
418 views

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 ...
2
votes
2answers
424 views

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 ...
0
votes
1answer
491 views

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 ...
2
votes
1answer
179 views

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. ...
1
vote
1answer
1k views

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 ...
1
vote
2answers
112 views

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 ...
3
votes
1answer
105 views

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 $1$. ...
0
votes
1answer
98 views

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?
-2
votes
1answer
274 views

Proof about cosets

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 $(i)$ ...
0
votes
1answer
79 views

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 ...
0
votes
0answers
334 views

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 ...
0
votes
3answers
454 views

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.
1
vote
1answer
104 views

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?
1
vote
2answers
128 views
1
vote
2answers
485 views

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, ...
0
votes
3answers
743 views

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 $N$?
1
vote
1answer
211 views

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 ...
3
votes
2answers
187 views

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, ...
1
vote
1answer
135 views

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 ...
0
votes
1answer
146 views

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 $z$ $$\cos(z)=\operatorname{Re}(\exp(iz))$$?
8
votes
1answer
864 views

“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 ...
1
vote
1answer
340 views

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 ...
2
votes
1answer
486 views

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 ...
0
votes
2answers
156 views

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
1
vote
1answer
349 views

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. ...
1
vote
1answer
70 views

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 ...
1
vote
1answer
77 views

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 ...
1
vote
0answers
97 views

About Lebesgue measure

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 ...
0
votes
0answers
132 views

Verifying a Limit Argument

I would like to verify the validity of the following argument. Let $f: \mathbb{R}^n/\left\{x_0\right\} \rightarrow \mathbb{R}^+$, $g: \mathbb{R}^n \rightarrow \mathbb{R}^+$ such that $0 \le f(x) \le ...
0
votes
1answer
223 views

Derivative of a Parameter Integral (involving a Gaussian density)

I have a question about the derivative of a certain parameter integral. To fix notation, let $\phi_{\mu,\sigma}(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp{\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right)}$ denote ...
5
votes
1answer
134 views

Bounds on an induced subgraph of a 4-critical graph

I just hate having to come here to ask questions. It's like accepting defeat that you can't solve it yourself.. but I've been trying to solve this for hours and can't find the intuition to solve the ...
0
votes
0answers
71 views

limit propert of Laplace transform

consider $f\in L^1(R_+)$ and define Laplace transform $$\mathcal{L}f(z):=\int_0^{\infty} f(s)e^{-zs}\mathbb{d}s. $$ How can I prove $$\lim_{\mathbf{Re}z\rightarrow\infty}\mathcal{L}f(z) = 0?$$ ...
0
votes
1answer
578 views

Squeeze/Sandwich Theorem in Two Variables

Suppose that $g,h : \mathbb{R} \rightarrow \mathbb{R}^+$ and $f: \mathbb{R}^2 / \left\{(0,0)\right\} \rightarrow \mathbb{R}^+$ and that $g(x_1) \le f(x_1,x_2) \le h(x_1)$ where not both $x_1,x_2$ are ...

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