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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3answers
86 views

Proof that if $x$ is an accumulation point, then $B(x,r)\cap A$ has infinite points

Let $(M,d)$ be a metric space, where $A\subset M$. If $x$ is an accumulation point of $A$, by definition $\forall r > 0,\; B(x,r)\setminus \{x\}\cap A \neq \emptyset$, so $\forall r > 0,\; ...
2
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1answer
69 views

Prove $\left|\frac{x^2y^3}{x^4+y^4}\right|\leq |x|+|y|$

How can we prove that $\left|\frac{x^2y^3}{x^4+y^4}\right|\leq |x|+|y|$? I got this as homework but don't even know where to start. I've tried developing $(x+y)^4$ but that didn't help to find a ...
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1answer
88 views
3
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1answer
67 views

Where is the mistake in the calculation of $y'$ if $ y = \Bigl( \dfrac{x^2+1}{x^2-1} \Bigr)^{1/4} $?

Plase take a look here. If $ y = \Bigl( \dfrac{x^2+1}{x^2-1} \Bigr)^{1/4} $ \begin{eqnarray} y'&=& \dfrac{1}{4} \Bigl( \dfrac{x^2+1}{x^2-1} \Bigr)^{-3/4} \left \{ \dfrac{2x(x^2-1) - ...
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1answer
86 views

Compute the following Integral

I want to compute the integral $$2\pi\int f(x) \sqrt{1+f'(x)^2} dx$$ where $f(x)=\dfrac{1}{e^x}$. I used maple and I found that the answer is: $$\pi e^{-2x} \left[e^{2x} ...
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2answers
406 views

Chromatic Polynomial

I am asked the following: Let n be a positive integer at least 3. The wheel W_n is the graph obtained by taking the cycle C_n, placing an additional vertex at the center, and joining it to ...
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2answers
129 views

$AA^{*}=I$ if and only if the rows of $A$ form an orthonormal basis

Suppose $A$ is an $n \times n$ matrix. Show that $AA^{*}=I$ if and only if the rows of $A$ form an orthonormal basis. So far the only thing that I have done with this problem is knowing that ...
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2answers
54 views

Subgroup of automorphism Group

Let $H \leq G$, $N := N_G(H) = \{g \in G\mid gHg^{-1} = H\}$ and $C:=C_G(H)= \{g \in G \mid \forall h \in H: ghg^{-1} = h\}$. I have shown that $C \lhd N$. How can I show that $N/C$ is isomorphic ...
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2answers
59 views

a problem in Lipschitz Functions

Prove that if $f$ is differentiable at $x_{0}$ then there exists a $ \delta>0 $ and a $K_{0}>0$ such that for all $x\in N_{\delta}(x_{0})$, $|f(x)-f(x_{0})|\leq K_{0}|x-x_{0}|$
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2answers
733 views

Find the antiderivative

How would I find the antiderivative of $f(t) = (3t^4-t^3+6t^2)/t^4$ Please show all of the steps that were done.
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2answers
250 views

Infinite series and logarithm

Is it true that: $$\log_e 2 = \frac12 + \frac {1}{1\cdot2\cdot3} + \frac {1}{3\cdot4\cdot5}+ \frac{1}{5\cdot6\cdot7}+ \ldots$$ It was one of my homeworks . Thanks!
1
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1answer
269 views

Surface area of a cylinder

I have been asked to find a parameterization for the surface $9=x^2+z^2,0\leq y\leq4$, and rewrite the surface integral $\iint y dS$ as a double integral. I believe that the parameterization should ...
2
votes
2answers
177 views

Finding angles of two vectors using simultaneous equations

In a physics problem, I am asked to find the resulting angle of two velocity vectors using each velocity vector's components. For the x-component, I have $m_av_{1ax} = m_av_{2ax} + m_bv_{2bx}$. ...
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2answers
42 views

Integral and Area of a section bounded by a function.

I'm having a really hard time grasping the concept of an integral/area of a region bounded a function. Let's use $x^3$ as our sample function. I understand the concept is to create an infinite ...
0
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2answers
99 views

probability( central limit theory)

A seed manufacturer sells seeds in packets of 50. Assume that each seed germinates with probability .99 independently of all the others.The manufacturer promises to replace, at no cost to the buyer, ...
2
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3answers
277 views

Conditional Expectation: What happens if you take conditional expectation on trivial sigma field?

Consider for the trvial $\sigma$ - field $\mathcal{F}_0 = \{\emptyset , \Omega\}$, What is Conditional expectation of the following in the following cases when $A = \emptyset$ and $A = \Omega$ ??? ? ...
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1answer
25 views

Weekly Compound Interest on $\$480$, $7\% $ interest, for $12$ years

So far I have the equation $480\left(1+\cfrac{0.07}{52}\right)^{52\times12}$ I think that is the right equation but it is not giving me the correct answer.
2
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2answers
577 views

Probability that the sum of all values of 5 pairs of dice will be between 30 and 40

I'm trying to solve a question that asks: If 5 pairs of fair dice are rolled, approximate the probability that the sum of the values obtained is between 30 and 40 inclusive. My approach so ...
0
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1answer
73 views

Poisson distribution and probability distributions

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
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1answer
176 views

Poisson distribution and probability of random variables

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
0
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1answer
29 views

Calculating the Probability of a random variable using a probability density function with integrals

Let $(\Omega, \mathcal F, P)$ be Lebesgue measeure on $[0,1]$, and set $X(\omega) = 1$ if $0 \le \omega < \frac{1}{4}$ $X(\omega) = 2\omega^2$ if $\frac{1}{4} \le \omega < \frac{3}{4}$ ...
1
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1answer
219 views

Using Lagrange multipliers for restricted extrema

Consider the function $f(x,y) = x^2 + xy + y^2$ defined on the unit disc $D = \{(x,y) \mid x^2 + y^2 \leq 1\}$. I can not simplify the equations to the point where I find a constant for the lagrange ...
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3answers
248 views

Is the optimal solution to this problem to row straight to the store?

For my homework, I was given this brainteaser: You’re sunbathing on the island shown on the map below. The island is six miles from shore at the closest point, and the nearest store is a ...
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1answer
322 views

Evaluating Logical Statements

The problem I am working on is, "Analyze the logical forms of the following statements: (a) Alice and Bob are not both in the room. (b) Alice and Bob are both not in the room. (c) Either Alice or ...
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3answers
1k views

How do I explain the Fundamental Theorem of Calculus to my teacher?

For extra credit for my class we are supposed to explain or describe to my teacher "The Fundamental Theorem of Calculus". Now I understand Calculus has a lot to do with integrals, differentiating, ...
5
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3answers
123 views

Evaluation of a definite integral.

In my real analysis course I was given this exercise: Calculate $\displaystyle{\int_0^1e^{x^2}dx}$. What I did was to write $\displaystyle{e^{x^2}=\sum_{n=0}^\infty\dfrac{x^{2n}}{n!}}$ and conclude ...
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1answer
38 views

calculating correlation

If one fair six-sided die is rolled, suppose that $X$ is the total number of even numbers shown and $Y$ is the total number of fives shown. How can I go about calculating the correlation exactly in ...
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1answer
915 views

Conditional probability question and solution

Please could someone review my solutions for the problems below..thanks in advance An e-mail message can travel through one of three server routes. The probability of transmission of error in each of ...
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1answer
173 views

Sine series of $\pi/2$

I'm studying Fourier series and came across this peculiar problem. I just studied (along with proper reasoning) that if $f(x)$ is an even function, then the fourier series has only Cosine terms and if ...
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4answers
1k views

What does it mean to say a random variable is non-negative?

How would you define a random variable to be non-negative ??? What are some examples of a Negative random variable ???
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1answer
31 views

Calculating the mean of a discrete R.V in a question

I have the following HW question: There are $N$ balls in a box, $m$ balls with an $S$ for success and $N-m$ balls with an $F$ for failure. Choose $n$ balls at random ($n\leq N$) and let $X$ = ...
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1answer
302 views

Discrete Mathematics, Graph Theory

Let $G$ be a connected graph in which every pair of edges has an endpoint in common. Show that $G$ is either a star or the complete graph $K(3)$.
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1answer
47 views

Problem related to the type of pde

I was trying to solve the following problem: The partial differential equation $y^{3}u_{xx}-(x^{2}-1)u_{yy}=0$ is (a) parabolic in $\{(x,y):x<0\}$, (b) hyperbolic in ...
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1answer
215 views

Derivatives and their domains

Derive the following functions and simplify as good as possible. Then determine the maximum domain (over $\mathbb{R}$) of each function $f$ and its derivative $f'$. $\displaystyle ...
2
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4answers
132 views

Center and Fitting subgroups

I have to prove that, if G is a finite soluble and nonabelian group, its center is a proper subgroup of the Fitting subgroup of G. In other words, that $Z(G)<F(G)$ Any ideas? Thanks a lot.
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1answer
92 views

Hypergeometric or?

I have a question regarding (a homework ) assignment. I've done some research but I couldn't get clear I were on the right track: $8$ people want to decide who is the designated driver. They each ...
1
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0answers
115 views

$\textrm{rank}(T)=\textrm{rank}(T^2)\implies \textrm{N}(T)\cap\textrm{R}(T)={0}$

Let $T:V\to V$ be a linear transformation from a finite dimensional vector space $V$. We are asked to show that $\textrm{rank}(T)=\textrm{rank}(T^2)\implies \textrm{N}(T)\cap\textrm{R}(T)={0}$. My ...
2
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1answer
97 views

a simple measure theory question (from homework)

Let X be a positive random variable independent of a standard Brownian motion B. Let $M_t = B_{tX}$ for t > 0. We assume that the random variable X is $F_t$ measurable for all t $\geq$ 0, require to ...
0
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2answers
280 views

Map is surjective/injective and a short exact sequence of groups

Im working on my thesis about semidirect products and splitting lemma. I got the following theorems to prove and Im a not sure how to start. I would appreciate any help. $\\$ 1. Let $f:A\to B$ be a ...
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1answer
205 views

Prove the following for $T$ a linear functional on $(X,\|·\|)$

(a) If $T$ is a bounded linear functional then $|T(f)|\leq \|T\|_*\|f\|$ for all $f\in X$. (b) $\|T\|_*=\sup\{|T(f)|\mid f\in X,\,\|f\|\leq 1\}=\sup\{|T(f)|\mid f\in X,\,\|f\|=1\}$ (c) A linear ...
0
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2answers
162 views

Why can we interchange summation sign and Variance?

Why can we interchange the summation sign and variance sign? $$\mathrm{Var}\left(\sum^n_{i=1} Y_i\right) = \sum^n_{i=1} \mathrm{Var}(Y_i) $$ Is there a proof for this?
2
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1answer
87 views

Linear relationship of a company's profit

Assume a linear relationship for a company that has several shops is not known. Let $Y_i$ be the profit the shop number $i$ makes in the coming year. Let $x_i$ be the size of the shop number ...
3
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2answers
265 views

Transitive action of normal subgroup of the alternating group

everyone! Would anyone be willing to give me any sort of help with the following question? Let $n\ge 4$ and $A_n$ the alternating group. Let $N$ a non-trivial normal subgroup of $A_n$. Prove that the ...
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2answers
116 views

Composite group homomorphism between alternating groups

Let $N$ a non-trivial normal subgroup of $A_n$ and $H = N \cap A_{n-1}$. I would like to show that $A_{n-1} \hookrightarrow A_n \to A_n/N$ is surjective, where $A_n \to A_n/N$ is the canonical ...
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0answers
94 views

On the centraliser of a normal subgroup and on group of automorphisms

The following question has been bothering me for sometime and any help will be greatly appreciated. Let me first fix notation. Let $G$ a group and for $a\in G$, let $\phi_a(x) = axa^{-1}, \forall x ...
3
votes
3answers
94 views

Criteria for metric on a set

Let $X$ be a set and $d: X \times X \to X$ be a function such that $d(a,b)=0$ if and only if $a=b$. Suppose further that $d(a,b) ≤ d(z,a)+d(z,b)$ for all $a,b,z \in X$. Show that $d$ is a metric ...
1
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1answer
57 views

Properly say superscripts locations

In english: $\bigl(x+y^2\bigr)$ is ('x' plus 'y' squared) $(x+y)^2$ is ('x' plus 'y' squared) How can I make the difference in english between the two?
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3answers
102 views

How to multiply out $(\sqrt{2}) (\sqrt{2}i)(\sqrt{2}+\sqrt{2}i)$

How to multiply out $(\sqrt{2}) (\sqrt{2}i)(\sqrt{2}+\sqrt{2}i)$? $i =$ the complex imaginary number.
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1answer
21 views

Distributions of random variables

Let $(\omega, F, P)$ be Lebesgue measeure on$[0,1]$, and set $X(\omega) = 1$ if $0 \le \omega < \frac{1}{4}$ $X(\omega) = 2\omega^2$ if $\frac{1}{4} \le \omega < \frac{3}{4}$ $X(\omega) = ...
1
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1answer
47 views

Metric Space - Accumulation

Let $(X,d)$ be a metric space. Let $A \subset X$ and $c \in X$. $c$ is called an accumulation point of $A$ if for every $\delta > 0$ there exists $a \in A$ such that $0 < d(a,c) < \delta$. ...