# 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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### Inequality - Find what value of $t$ satisfies: $(t/24) - (t+1) + (3t/8) < (5/12) (t+1)$

Inequality - Find what value of $t$ satisfies: $(t/24) - (t+1) + (3t/8) < (5/12) (t+1)$. Step 1: I multiplied both sides by $24$ and divided to get: $t-24(t+1)+9t < 10+24(t+1)$. Step 2: I ...
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### Free action by cyclic group.

Let $G$ be a group acting on a set $X$. If $g\in G$ has no fixed points, prove or disprove the cyclic group $\left \langle g \right \rangle$ acts freely on $X$. edit: Can also assume $g$ has finite ...
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### To derive the weak of this fem?

It was long ago I studied FEM, and now, I am stucked in this exercise. $$-\Delta u + u = f \text{ in } \Omega \\ u = 0 \text{ on } \partial \Omega$$ where the domain ...
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### Finding cut-off point for utility function

OK, so apologies for the easy question, but I'm new to this! This is somewhere between elementary algebra, and beginner's game theory. The question comes from a paper I read here (see p. 193): ...
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### Need a solution to find the locus of an equation. [on hold]

Find the equation of the locus of a point which moves so that its distance from $(a,0)$ is equal to its distance from the $y$-axis. The answer is $$y^2 - 2ax + a^2 =0$$ Please can someone find the ...
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### Complicated Logic Proof involving Tautology and Law of Excluded Middle

I'm having great difficulty solving the following problem, and even figuring out where to start with the proof. $$\neg A\lor\neg(\neg B\land(\neg A\lor B))$$ Please see the following examples of ...
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### Why integers are not enough? [on hold]

My Maths teacher asks me this in my homework and i cant find it anywhere. Thanks for your help. P.S.: she asks another thing... ''why rational numbers are not enough?'' could you help me with that ...
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### Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. (weird equation)

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five ...
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### What did I do wrong?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
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### Linear Programming to find the loan plan to minimize the interest payment

Assume that it is the first of July and you are running a small shop. The sales revenue and the amount of bills you have to pay for the next six months are estimated as following: In short, you ...
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### Proof that $1+4^n+7^n$ is divisible by 3 for all $n \in \mathbb{N}$

So, as the title said I have to proof that $1+4^n+7^n$ is divisible by 3 for all $n \in \mathbb{N}$. I have to do it with induction. So I got my start, for $n=0$: We have $1+4^0+7^0 = 1+1+1 = 3$ and ...
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### Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
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### An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
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I have $$J_{\theta}(X) = - \frac 1 m \cdot \left[ y \cdot ln( h_{\theta} (X ) ) + ( 1 - y) \cdot ln ( 1 - h_{\theta}(X) ) \right]$$ I need $\frac d {d\theta} J_{\theta}(X)$. I tried many time, and ...
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### How can i find $dz$ and $d^2(z)$ from implicit equation?

Given the function $F(x+z,y+z)=0$,i need to find $dz$ and $d^2z$....here's my attempt: $$dF=F'_1d(x+z)+F'_2d(y+z)=0$$ $$F'_1(x'dx+z'dz)+F'_2(y'dy+z'dz)=0$$ $$F'_1(dx+dz)+F'_2(dy+dz)=0$$ ...
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### Points defined by relations (an exercise from “System of Coordinates”)?

An exercise from "System of Coordinates" (by Gelfand, Glagoleva and Kirilov) asks me to "[t]ry to decide by yourself which sets of points are defined by these relations" and relations given are: a. ...
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### Related rates of change: distance between a plane and a car

A highway patrol plane flies $3 m$ above a level, straight road at a constant speed of $120 m/s$. The pilot sees an oncoming car and with radar determines that at the instant the line of sight ...
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### Linear algebraic group

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
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### Copy of C in H , trace is independent of the choice

Let X€ Mn(H). For each of the choices of a copy of C in H , write out the corresponding matrix of X as an element of M(2n,C). Use this formula to show that the trace of X is independent of the choice. ...
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### On modular group and quadratic forms

Let $\Gamma$ be the modular group, is the group of linear transformations of the upper half of the complex plane. Let $\mathbb Q_{N^2{d_K}}/\Gamma$ (the group of positive definite primitive quadratic ...
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### An application of squeeze theorem to the limit of $(\sin x)/x$ [duplicate]

How can I solve the following problem using elementary trigonometry? Use the Squeeze Theorem to conclude that $$\lim_{x\to 0^+} \frac{\sin(x)}{x} =1.$$ Use symmetries of $y = \frac{\sin(x)}{x}$ to ...
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### Real inner product from a complex one

Let $V$ be a complex vector space. We may view $V$ as a real vector space by simply ignoring non-real scalars. Now suppose that $\langle \cdot,\cdot,\rangle$ is a complex inner product on $V$, and ...
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### I know the perimeter of the rectangle but not the area. How do I find the length and width?

The perimeter of the rectangle is 986. I don't know the area and I need to find the length and width. The problem states that the length is 199 ft more than the width. That is all the information that ...
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### How to solve this: $|3-x|\ge2$

How to solve $|3-x|\ge2$ ? I know that if $|x| < y$, then $-y < x < y$. But in this case what to do? Thanks. Here, $|x|$ is the absolute value of $x$.
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### Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace$ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
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### How to show the cyclotomic polynomial is irreducible over $\mathbb{R}(T,\sqrt[n]{T})$

I'm trying to solve the following problem. Let $T$ be a transcendental over $\mathbb{R}$. Put $K=\mathbb{R}(T), n\geq3$. Let $L$ be the least splitting field of $X^n-T$. Then, calculate $[L:K]$. I ...
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### Determining Moving-Average Representation of AR(2) Process

Consider a stationary $AR(2)$ process given by $$X_{t} - X_{t-1} + 0.25X_{t-2} = 5 + a_{t}$$ where $a_{t} \sim WN(0,1)$ (white noise). I am interested in obtaining the causal representation of ...
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### Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
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The path of a football flying through the air can be modelled by a quadratic equation. The football reaches the ground after 12 seconds in flight and is kicked from a height of 1 meter. The parabola ...
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### Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
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### exercise on $L(0,1)$ and functional analysis

how could i prove that $u_{k}(x)=sin(kx) \to 0$ as $x\to \infty$ in $L^{2}(0,1)$ how could i prove that for fixed a,b real numbers and $0 < \lambda <1$ and for the function defined as ...
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### How to compute the length of line segments inside an isosceles trapezoid parallel to the base

here is the problem in a picture, sorry for the drawing. i have found how to compute the midline of the isosceles trapezoid, 1/2 (top+bottom) , but there seems to be no formula for other line segments ...
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### Equation of horizontal/vertical line and changing to $y=mx+c$ format

I've been given the equation $2x-3y=5$. I was wondering whether this is a horizontal or vertical equation and how would I rearrange this to $y=mx+c$. I know that this is a fairly basic equation but ...
Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...