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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3
votes
1answer
167 views

Selecting books from a shelf

A shelf contains 24 books. How many ways can 6 books be selected from these 24 with the restriction that no two selected books can be adjacent? So first we want to divide by 2 to fulfill the adjacent ...
1
vote
2answers
137 views

Calculus, integration, Riemann sum help?

Express as a definite integral and then evaluate the limit of the Riemann sum lim $$ \lim_{n\to \infty}\sum_{i=0}^{n-1} (3x_i^2 + 1)\Delta x, $$ where $P$ is the partition with $$ x_i = -1 + ...
0
votes
0answers
410 views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
1
vote
0answers
60 views

Analogue for finite sums of $$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) (*)$$

Please help me to find an analogue for finite sums of $$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) (*)$$ where $ \bar{f}(t)=\frac{1}{t}\int_{a}^{t} ...
1
vote
1answer
43 views

Difference Between Random Variable and Fraction

Let $\displaystyle \epsilon > 0$ and $\displaystyle X$ be uniformly distributed on $[0,1]$. Prove that, almost surely, there exists only a finite number of rationals $\displaystyle \frac{p}{q}$, ...
12
votes
2answers
310 views

6 point lying on a common circle

$Z$ is an interior point of segment $XY$. Three semicircles are drawn over segments $XY$, $XZ$ and $ZY$ on the same side. The midpoints of the arcs are $M1$, $M2$ and $M3$ respectively. A circle ...
4
votes
1answer
155 views

Homeomorphism theorem

I need to show that if $f: X \rightarrow Y$ is 1-1 and $X$ and $Y$ are metric spaces, then if $\forall A\subset X, f(\overline{A})=\overline{f(A)} $ then $f$ is homeomorphism. 1) Assume $f$ is 1-1 ...
4
votes
2answers
1k views

Convergence of $\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$

Please suggest some hint to test the convergence of the following series $$\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$$
0
votes
0answers
175 views

Implementing the $\theta$-method for solving an IVP

I want to implement the $\theta$-method to solve an IVP in MATLAB. The $\theta$-method is: $y_{j+1} = y_j + h[\theta f(t_j, y_j) + (1 - \theta)f(t_{j+1}, y_{j+1})]$ for $\theta \in [0, 1]$. I want ...
0
votes
1answer
94 views

Number of $k$-cycles in permutations of $[2k]$?

What is the expectation of the number of $k$-cycles in a randomly selected permutation of $[2k] = {1,2, . . . ,2k}$?
2
votes
0answers
33 views

Is the answer given to Calculus 9e by Larson section 9.5 question 37.

The problem Approximate the sum of the series by using the first six terms. question #37) $\sum \limits_{n=0}^{\infty} \dfrac{\left(-1\right)^n 2}{n!}$ The answer given by CalcChat*: ...
0
votes
2answers
79 views

How to show that $f-g$ is imaginary constant in $\mathbb{D}$?

How to show that $f-g$ is imaginary constant in $\mathbb{D}$? Let $f$ and $g$ be continuous functions in $\bar{\mathbb{D}}$ and analytic in $\mathbb{D}$. Show that if $\mathfrak{R}f=\mathfrak{R}g$ at ...
3
votes
3answers
105 views

If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$, then $M$ has the structure of an $R/I$-module

Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: ...
0
votes
1answer
923 views

Finding Mean Value and Standard Deviation

The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than ...
0
votes
2answers
54 views

how prove $\sum_{i=1}^{k}|n_i-m_i|$ is even such that $m_1$,$m_2$,…,$m_k$ is permutation of $n_1$,$n_2$ ,…,$n_k$?

let $n_1$,$n_2$ ,...,$n_k$ be arbitrary integer and $m_1$,$m_2$,...,$m_k$ is permutation of $n_1$,$n_2$ ,...,$n_k$ how prove $$|n_1-m_1|+|n_2-m_2|+|n_3- m_3|+...+|n_k-m_k|$$ is even numbers ? ...
0
votes
1answer
64 views

Tell if $S = (X, \Sigma)$ is a distributive complemented lattice

Let $\Sigma$ be a partial order defined over $\mathbb N^* = \mathbb N \setminus \{0\}$ such that $$n\;\Sigma\;m \Leftrightarrow n=m \text{ or } \text{r}(8,n) \text { is a non-trivial divisor of } ...
0
votes
2answers
240 views

Vertex Cover degree problem

So Vertex cover (VC): Instance: a graph $G$ and an integer $k>0$. Question: Does $G$ have a vertex cover of size at most $k$? We will now define a version of this problem in which we assume that ...
2
votes
2answers
53 views

Finding the number of $2$-element chains of $B_n$

I'm trying to calculate the number of $2$-element chains and anti-chains of $B_n$, where $B_n$ is the boolean algebra partially ordered set of degree $n$. I understand that I want to take all of the ...
2
votes
1answer
67 views

Diameter of graph is $\geq 4$

Let $G$ be any graph such that $\Delta G=k\geq2$. If there are at least $k^3-k^2+k+2$ vertices, show that $\operatorname{diam}(G)\geq 4$. Can any one give idea on how to approach this problem
2
votes
2answers
559 views

Integral of $\int \frac {\sqrt {x^2 - 4}}{x} dx$

I am trying to find $$\int \frac {\sqrt {x^2 - 4}}{x} dx$$ I make $x = 2 \sec\theta$ $$\int \frac {\sqrt {4(\sec^2 \theta - 1)}}{x} dx$$ $$\int \frac {\sqrt {4\tan^2 \theta}}{x} dx$$ $$\int \frac ...
2
votes
2answers
78 views

What function satisfies these conditions?

In order to solve the PDE I am working with, I need to determine a function $u(x,t)$ that satisfies both of these conditions $$u(0,t)=\sin(t)$$ $$u(1,t)=\cos(t)$$ I know it is just trial and error ...
2
votes
1answer
108 views

how to solve this nonlinear equation system by changing this system to linear system

consider following nonlinear equation system how solve it? $$x'=|y|$$ $$y'=x$$ and whats the matrix that associated to this system
1
vote
0answers
109 views

Showing that a Boolean algebra is a Boolean ring

I've proved that a Boolean ring is a Boolean algebra but I am having trouble with the converse. The operation for + is defined as the symmetric difference for elements $a$ and $b$ from the Boolean ...
0
votes
1answer
144 views

Finding Confidence Intervals of Variants of Experimental Means of Normal Distributions

Let $X_1, \dots, X_n$ be mutually independent random variables such that $X_j \sim N(j\theta, \sigma^2)$ for $j = 1,\dots, n \ $ where $\ \sigma^2 > 0 \ $ is known. $$ \bar{X} = \frac{\sum^n_{j = ...
2
votes
1answer
145 views

Proving some properties of pointwise convergent sequences.

It is given that $f_n \rightarrow f$ pointwisely on a domain $D$ and $g_n\rightarrow g$ pointwisely on $D$. I'm trying to prove these two properties: $f_ng_n\rightarrow fg$ pointwisely on $D$ ...
1
vote
3answers
150 views

Finding the $x$ and $y$ values such that the partial derivatives are zero simultaneously

$f(x,y) = x^2 + 4xy + y^2 -4x + 16y + 3$ So, I proceeded with taking the partial derivatives: $f(x,y)_x = 2x + 4y - 4$ and $f(x,y)_y = 4x + 2y + 16$ and $f(x,y)_x = f(x,y)_y = 0$ $2x + 4y - 4 = ...
1
vote
3answers
109 views

Differentiation operator and eigenvalues

Let $V = \{p(x) \in F[x] \ | \ \deg(p(x)) \le n\}$. Let $T : V \to V$ be given by differentiation, in essence $$T(p(x)) = p'(x)$$ It seems to me that the only eigenvalue that can exist is $\lambda ...
0
votes
1answer
59 views

Differential Equation, phase curve.

Let $x:I \rightarrow \mathbf{R}^n$ be a solution defined on an interval $I \subset \mathbf{R} $. Consider the ODEs $$ \dot{x} = x, $$ $$ \dot{y} = ky, $$ where $k$ is a constant. Is the curve given ...
3
votes
2answers
195 views

Translation of : The disjunction of two contingencies can be a tautology.

The statement is: "The disjunction of two contingencies can be a tautology." The predicates are: $C(x)$: "$x$ is a contradiction." $T(x)$: "$x$ is a tautology." The book says the answer is ...
2
votes
3answers
1k views

Prove that the rank of a block diagonal matrix equals the sum of the ranks of the matrices that are the main diagonal blocks.

\begin{equation*} X= \begin{pmatrix} A& 0 \newline 0& B \end{pmatrix} \end{equation*} If A and B are some matrices and 0 is a zero matrix, prove that $\ rank(X)=\ rank(A)+\ rank(B)$. Also, if ...
0
votes
2answers
65 views

probability of a coin flip

A fair coin if flipped 3 times. If (F1, F2, F3) denotes a typical flip sequence, let E1 denote the event that at least two of the Fi's are Heads, let E2 denote the event that exactly two of the Fi's ...
2
votes
3answers
3k views

probability selecting marbles

can someone solve this example? An urn contains 2 Red marbles, 3 White marbles and 4 Blue marbles. You reach in and draw out 3 marbles at random (without replacement). What is the probability that ...
4
votes
1answer
72 views

Points on a plane

I have been assigned this problem and am not sure how to approach it! Please help me figure out what I should do! Let $S$ be a finite set of points in a plane chosen to have the property that for ...
1
vote
1answer
441 views

Obscure Probability Question

Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and standard deviation 5 (information in the article “Mathematical Model of Chloride Concentration in ...
3
votes
1answer
45 views

An equation in tensor product

I want to find the numbers of solutions of below equation:$$| \mathbb Z_n \otimes \mathbb Z_{12}| =\frac{n}{2} $$
2
votes
3answers
48 views

Show $I=\{sf\mid s\in \mathbb Z_{11}[x]\}$ matches $J=\{h \in \mathbb Z_{11}[x] \mid h(1) = h(-2) = 0\}$.

Let $f=x^2+x+\overline{9} \in \mathbb Z_{11}[x]$. Show that $I=\left\{sf\mid s\in \mathbb Z_{11}[x] \right\}$ matches $J=\left\{h \in \mathbb Z_{11}[x] \mid h(\overline{1}) = h(\overline{-2}) = ...
-1
votes
2answers
173 views

Give an example of a group of order 8 in which every non-identity element has order 2

This is just an optional question in my homework. It follows the question proving groups with all elements order 2 have to be Abelian. I have done that but I couldnt come up with an example for a ...
2
votes
2answers
115 views

Uniform convergence of $f_n(x)=\frac{n^2\sin{(nx)}}{1+n^2x}$ on $(0,\infty)$

I'm trying to first check if $(f_n)$ is pointwise convergent here but I seem to be having an issue with that much. I was trying to split this into $x$ either being a multiple of $\pi$ or not. But ...
0
votes
2answers
53 views

Uniform convergence of a certain series

I want to check the uniform convergence of $\sum (1-x)x^k$ and $\sum (1-x)^2x^k$ for $0 \leq x <1$ My attempt, $\sum (1-x)x^k=(1-x)\sum x^k$ , $\sum (1-x)^2x=(1-x)^2\sum x^k$ and $\sum x^k $ is ...
3
votes
2answers
667 views

Damped Harmonic Oscillator [ Math not physics question]

Question Consider, The motion of a a damped harmonic oscillator is described by $x^{''}$ + $bx^{'}$ + $kx = 0$, where b is greater than or equal to 0 A) Rewrite as a two dimensional linear system. ...
0
votes
2answers
98 views

Determine if $f_n(x)=x^n\sin{(nx)}$ is uniform convergent on the following intervals.

The intervals are: $I_1=\{ x\in\mathbb{R}:a\leq x\leq b\}$ with $-1<a<b<1$. $I_2=\{ x\in\mathbb{R}:-1< x< 1\}$. I've shown $f_n$ is pointwise convergent to $0$ on $I_1$ and $I_2$. ...
2
votes
1answer
63 views

need help Jordan base

Need help, how to find Jordan base for matrix: $A=\begin{pmatrix} -1&-1 &-2 &4 \\ 1&-3 &1 &-2 \\ 0&0&2&-8\\ 0&0 & 2&-6 \end{pmatrix}$ I found the ...
1
vote
1answer
70 views

Showing $f$ is homeomorphism

Let $(X,d)$ be metric space: Let $f$ be an isometry on $X$ Let $f(X)$ be dense is $X$. Then how do I show that $f$ is a homeomorphism. I have shown $f$ to be one-one. Let $f(x)=f(y)$. This ...
2
votes
1answer
105 views

Can someone explain the steps needed to set up this equation to solve with Simpson's rule?

A fleeing object leaves the origin and moves up the $y$-axis. At the same time, a pursuer leaves the point $(1,0)$ and moves always toward the fleeing object. If the pursuer's speed is twice that of ...
5
votes
2answers
2k views

Cauchy product of two absolutely convergent series is absolutely convergent. (Rudin PMA Ch. 3 ex 13)

Given $\sum a_n$ and $\sum b_n$ we define the Cauchy product to be $\sum_{k = 0}^{n}a_kb_{n-k}$. I need to prove that if both $\sum a_n$ and $\sum b_n$ are absolutely convergent then so is the ...
0
votes
0answers
478 views

Hypothesis Testing on Exponential distributions

Let $X_1, \dots, X_n$ be independent exponential $(\theta)$ random variables. Suppose we are interested in testing $H_0: \theta = \theta_0 = 1$ versus $H_A: \theta = \theta_1>1$. Consider two tests ...
4
votes
1answer
65 views

Find separable irreducible $g$ such that $f(x)=g(x^{p^d})$

This is an exercise from VII.4. in Algebra: Chapter 0. Let $\mathcal{k}$ be a field of characteristic $p$, and $f(x)\in\mathcal{k}[x]$ an inseparable irreducible polynomial. Find a separable ...
1
vote
1answer
418 views

Finding the x and y values such that the partial derivatives are zero simultaneously (part two)

$f(x,y)= \ln(x^2 + y^2 +1)$ The partial derivatives of $f(x,y)$ being: $f_x(x,y) = \frac{2x}{x^2+y^2+1}$ and $f_y(x,y)=\frac{2y}{x^2+y^2+1}$ Setting each partial derivative equal to zero, adding ...
1
vote
1answer
368 views

Rao-Blackwell Uniform Distribution

I am having a bit of an argument with my study group about a Rao-Blackwell problem that we have for our statistical theory class. The problem goes like this: Let X~U(0,$\theta$), and suppose we have ...
2
votes
3answers
82 views

$\lim_{n\to\infty}\dfrac{1}{n}\sum_{k=0}^n\cos\dfrac{\pi k}{2n}=\dfrac{2}{\pi}$

Please help me to show that $$\lim_{n\to\infty}\dfrac{1}{n}\sum_{k=0}^n\cos\dfrac{\pi k}{2n}=\dfrac{2}{\pi}$$ I'm absolutely clueless. A hint rather than a complete solution would be more ...