# 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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### A question on a continuous, nonegative function

Let $f$ be a continuous, nonegative function and $\alpha$ a root of $f$. I want to find a function $g$ such that $\lim_{x\to \alpha}f(x)g(x)=\infty$ and $\lim_{x\to \beta}f(x)g(x)=0$ for every root ...
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### Prove or Disprove the following statement. For any sets $A$, $B$, and $C$, we have $A \cup (B \& C) = (A\cup B) \cup (A\&C)$

Trying to figure this question out in my proofs class (tried venn-diagram the multiple set-notation signs are confusing me). Homework question in the fundamental sets unit.
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### Queue system with queue-triggered input process

I have a queue system, a classic system with an input generator, a queue and a servant. The servant is a $M$-servant with a certain serving rate $\mu$. The queue can contain an infinite number of ...
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### Find characteristic equation of transfer function

G(s) = K/[s(s+1)(s+5)] I simplify G(s) into K/(s^3+6s^2+5s). So is the characteristic equation s^3+6s^2+5s = 0 ? What does it exactly mean? I'm sometimes confused because I've personally memorized ...
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### Statistics about unbiased estimator with an unknown parameter

Way too hard questions for me. Could anyone give me some clues or way to solve?
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### How to prove this equation of composition of binary relations?

We have three binary relations $R,S,T$ all from $A$ to $A$. Prove $(R∘S)\cap T=\emptyset$ iff $(R^{-1}∘T)\cap S=\emptyset$ iff $(T∘S^{-1})\cap R=\emptyset$ I don't quite see what's $(R∘S)^{c}$ is ...
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### Volume of oblique cylinder

In my homework, was given an oblique cylinder with a radius of 3, slant height of 6, a slant angle of 60 degrees. I do not understand how to find volume of slanted cylinders. Can anyone help me ...
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### Understanding for a system why not asymptotically stable when $\epsilon = 0$ at the origin

Consider the system $$x' = (\epsilon x+2y)(z+1)$$ $$y' = (-x+\epsilon y)(z+1)$$ $$z' = -z^3$$ Show that the origin is not asymptotically stable when $\epsilon = 0$. I am told if we start with ...
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### Volume of revolution of cardioid

The parametric equations of a cardioid are $x=\cos\theta (1-\cos\theta)$ and $y=\sin\theta (1-\cos\theta)$, $0\le\theta\le 2\pi$. Diagram here. The region enclosed by the cardioid is rotated about the ...
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### a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
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### Discrete and compact subset must be finite

Show that a discrete and compact subset $D \subset \mathbb{C}$ must be finite. Does this conclusion hold if $D$ is just discrete and bounded? How about discrete and closed? Compact is the usual (for ...
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### Clarify my understanding for central limit theorem from a statement

Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student ...
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### Borel measurable function

I'm struggling on the following question from a past paper: Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function and let $h:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined ...
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### conditional probability - implication of fraction

$A,B,C$ random variables with $p(A) = p(A\mid C)$ for the pdfs, so $A$ and $C$ are independent. When does the following hold: $p(A\mid B,C) = p(A\mid B)$? Would it additionally also require $B$ and ...
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### Linear Algebra Questions over vector Spaces

I have 3 questions and I am wondering whether or not the following are always true or never true. 1.If the null space of 5x4matrix is 2D, then the column space of the matrix can be isomorphic to a ...
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### Composition Series of $A_4 \times S_5$

Please help me with the following question: Find the composition series of $A_4 \times S_5$ and prove that this series is indeed a composition series. Afterwards, find a group with the same ...
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### Time-Scale Decomposition: The slow variable only has a single unstable steady state?

I'm completing a time-scale decomposition of a set of equations. The original set of two equations is $$S'= 3k_1S^3E_T + (3k_{-1}-3k_1S^3)E_1$$ $$E_1'=-k_1S^3E_T+(k_{-1}+k_2+k_1S^3)E_1$$ I was ...
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### Given a solution flow to find periodic solutions

Given the system of differential equations $x' = 2x + y^3$ and $y' = -y$ i found the flow $$\phi_t(x,y) = ((x_0 + 1/5y_0^3)e^{2t} - 1/5 y_0^3e^{-3t}, y_0 e^{-t})$$. I am wondering are there any ...
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### Finding the matrix associated with a linear transformation

I've been staring at this problem for over an hour, trying different things and getting approximately nowhere. Here's the problem: Let ...
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### Prove that the series converges

Let $(a_n)$ be a bounded positive and monotone increasing sequence. I need to show that $\sum (1-\frac{a_n}{a_{n+1}})$ converges. My approach was as follows: Let $B=sup (a_n)$ Then since the ...
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### How do I find the Jacobi matrix?

I've never done questions like these, so I would very much like some help. We are given a function $f: \mathbb R^n \to \mathbb R$ given by $f(x)=\langle x,\xi\rangle^2$ where $\langle\,,\rangle$ is ...
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### Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $K[X_1,X_2,X_3,X_4]$

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=K[X_1,X_2,X_3,X_4]$?
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### Discrete subsets in real analysis

So this is the first time I have seen the definition for a "discrete" subset. I want to first make sure that my understanding is correct. If $D$ is discrete, then D is essentially just the boundary ...
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### Distributing $n$ different things among $r$ persons

How can $10$ different pencils be distributed among $3$ students? MY TRY $1$ total ways $= 3^{10}$ MY TRY $2$ $10 \times 9 \times 8 =720$ Which one is correct? If both are wrong what is correct ...
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### Poisson Process and Conditional Probability

Let $X= (X(t); t\ge0)$ be a poisson process with the intensity ($\lambda$ per hour) A) Find the conditional probability of having $m$ events in the first $t$ hours, given that there were $n$ events ...
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### Problems understanding error of approximation

I am to use Corrollary C to find a better approximation for my problem. I'm trying to understand what exactly is going on but I'm having severe problems understanding it. I sort of understand ...
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### Binomial series property for $|z| \lt 1$

I had a lecture where we proved the following identity: $\forall z \in \mathbb{C}$ with $|z| \lt 1$. $$\frac{1}{(1-z)^{\alpha}}=\sum_{k=0}^\infty {\alpha +k-1\choose k}z^k$$ Now i know this proof is ...