# 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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### Descartes rule of signs extension

Let $V(\text{sequence})$ be the number of sign changes in the sequence, e.g. $V(-3,0,-2,9,0,1)=1$. Show that $V(a_0,a_1,...,a_n)\ge V(a_0,a_0+a_1,a_0+a_1+a_2,...)$. Furthermore, prove that if ...
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### Invertible Matrices Proof

Given that B is an invertible matrix and $B^3 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$. (where $I$ is an identity matrix) $B$ is a matrix that is $n \times n$.
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### Real Analysis: Showing $f: \Bbb Q \to \Bbb Q$ is continuous

The following is all working in $\mathbb{Q}$, not $\mathbb{R}$. I am working with the function $f: \mathbb{Q} \to \mathbb{Q}$ defined piece-wise by $f(x)=-1$ if $x^2<2$ $f(x)=1$ if otherwise I ...
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### Prove that if Triangles ABC = DEF in a metric geometry, then line AB contains exactly two of the points D, E, and F.

Prove that if Triangles ABC = DEF in a metric geometry, then line AB contains exactly two of the points D, E, and F. We are not allowed to use the facts: In a metric geometry, if triangles ABC=DEF, ...
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### Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
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### Proof that arithmetic mean is greater than geometric mean? [duplicate]

I have to prove that $\frac{x + y}{2}> \sqrt{xy}$ algebraically for any $x,y \in \mathbb{R}$ such that $x,y \ge 0$ and $x\ne y.$ I'm fairly confused as to how to solve this problem algebraically, ...
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### How to Prove This Trigonometry Identity?

I have to prove that: $$\tan^2\theta \sin^2\theta = \tan^2\theta - \sin^2 \theta$$ Here is what I have tried $$\tan^2\theta \sin^2\theta$$ ...
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### Probablity that 3 husbands sit next to their wives round a circular table

There are 3 couples sitting randomly round a 6-seater circular table. What is the probability that all the husbands and wives sit next to each other? My attempt: First wife, say, takes any of the ...
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### Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
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### Binary relation, reflexive, symmetric and transitive

I have a question regarding an image. I'm currently studying binary relations and the following image confused me: What got me confused is that the page from which I got the link ...
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### Help finishing this exercise!

Given the following functions: $$F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $F$ is ...