# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 $D$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
I want to find the numbers of solutions of below equation:$$| \mathbb Z_n \otimes \mathbb Z_{12}| =\frac{n}{2}$$
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 $J=\left\{h \in \mathbb Z_{11}[x] \mid h(\overline{1}) = h(\overline{-2}) = ... 2answers 173 views ### Give an example of a group of order 8 in which every non-identity element has order 2 This is just an optional question in my homework. It follows the question proving groups with all elements order 2 have to be Abelian. I have done that but I couldnt come up with an example for a ... 2answers 115 views ### Uniform convergence of$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$or not. But ... 2answers 53 views ### Uniform convergence of a certain series I want to check the uniform convergence of$\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 $is ... 2answers 667 views ### Damped Harmonic Oscillator [ Math not physics question] Question Consider, The motion of a a damped harmonic oscillator is described by$x^{''}$+$bx^{'}$+$kx = 0$, where b is greater than or equal to 0 A) Rewrite as a two dimensional linear system. ... 2answers 98 views ### Determine if$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$I_2$. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ...